Algebra Examples

$\begin{array}{c} x & f (x) \\ 216 & 3 \\ 1296 & 4 \\ 7776 & 5 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.2

Build a set of equations from the table such that $f (x) = a x + b$ .

$3 = a (216) + b 4 = a (1296) + b 5 = a (7776) + b$

Step 1.3

Calculate the values of $a$ and $b$ .

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Step 1.3.1

Solve for $b$ in $3 = a (216) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (216) + b = 3$ .

$a (216) + b = 3$

$4 = a (1296) + b$

$5 = a (7776) + b$

Step 1.3.1.2

Move $216$ to the left of $a$ .

$216 a + b = 3$

$4 = a (1296) + b$

$5 = a (7776) + b$

Step 1.3.1.3

Subtract $216 a$ from both sides of the equation.

$b = 3 - 216 a$

$4 = a (1296) + b$

$5 = a (7776) + b$

$b = 3 - 216 a$

$4 = a (1296) + b$

$5 = a (7776) + b$

Step 1.3.2

Replace all occurrences of $b$ with $3 - 216 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $4 = a (1296) + b$ with $3 - 216 a$ .

$4 = a (1296) + 3 - 216 a$

$b = 3 - 216 a$

$5 = a (7776) + b$

Step 1.3.2.2

Simplify $4 = a (1296) + 3 - 216 a$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$4 = a (1296) + 3 - 216 a$

$b = 3 - 216 a$

$5 = a (7776) + b$

$4 = a (1296) + 3 - 216 a$

$b = 3 - 216 a$

$5 = a (7776) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (1296) + 3 - 216 a$ .

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Step 1.3.2.2.2.1.1

Move $1296$ to the left of $a$ .

$4 = 1296 a + 3 - 216 a$

$b = 3 - 216 a$

$5 = a (7776) + b$

Step 1.3.2.2.2.1.2

Subtract $216 a$ from $1296 a$ .

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = a (7776) + b$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = a (7776) + b$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = a (7776) + b$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = a (7776) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $5 = a (7776) + b$ with $3 - 216 a$ .

$5 = a (7776) + 3 - 216 a$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.2.4

Simplify $5 = a (7776) + 3 - 216 a$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$5 = a (7776) + 3 - 216 a$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = a (7776) + 3 - 216 a$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (7776) + 3 - 216 a$ .

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Step 1.3.2.4.2.1.1

Move $7776$ to the left of $a$ .

$5 = 7776 a + 3 - 216 a$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.2.4.2.1.2

Subtract $216 a$ from $7776 a$ .

$5 = 7560 a + 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = 7560 a + 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = 7560 a + 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = 7560 a + 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$5 = 7560 a + 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3

Solve for $a$ in $5 = 7560 a + 3$ .

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Step 1.3.3.1

Rewrite the equation as $7560 a + 3 = 5$ .

$7560 a + 3 = 5$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 1.3.3.2.1

Subtract $3$ from both sides of the equation.

$7560 a = 5 - 3$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.2.2

Subtract $3$ from $5$ .

$7560 a = 2$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$7560 a = 2$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3

Divide each term in $7560 a = 2$ by $7560$ and simplify.

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Step 1.3.3.3.1

Divide each term in $7560 a = 2$ by $7560$ .

$\frac{7560 a}{7560} = \frac{2}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $7560$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{7560 a}{7560} = \frac{2}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{2}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{2}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Cancel the common factor of $2$ and $7560$ .

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Step 1.3.3.3.3.1.1

Factor $2$ out of $2$ .

$a = \frac{2 (1)}{7560}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.3.1.2

Cancel the common factors.

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Step 1.3.3.3.3.1.2.1

Factor $2$ out of $7560$ .

$a = \frac{2 \cdot 1}{2 \cdot 3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot 1}{2 \cdot 3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

$a = \frac{1}{3780}$

$4 = 1080 a + 3$

$b = 3 - 216 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{1}{3780}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $4 = 1080 a + 3$ with $\frac{1}{3780}$ .

$4 = 1080 (\frac{1}{3780}) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $1080 (\frac{1}{3780}) + 3$ .

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Step 1.3.4.2.1.1

Simplify each term.

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Step 1.3.4.2.1.1.1

Cancel the common factor of $540$ .

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Step 1.3.4.2.1.1.1.1

Factor $540$ out of $1080$ .

$4 = 540 (2) (\frac{1}{3780}) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.1.1.2

Factor $540$ out of $3780$ .

$4 = 540 \cdot (2 (\frac{1}{540 \cdot 7})) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.1.1.3

Cancel the common factor.

$4 = 540 \cdot (2 (\frac{1}{540 \cdot 7})) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.1.1.4

Rewrite the expression.

$4 = 2 (\frac{1}{7}) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

$4 = 2 (\frac{1}{7}) + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.1.2

Combine $2$ and $\frac{1}{7}$ .

$4 = \frac{2}{7} + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

$4 = \frac{2}{7} + 3$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$4 = \frac{2}{7} + 3 \cdot \frac{7}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.3

Combine $3$ and $\frac{7}{7}$ .

$4 = \frac{2}{7} + \frac{3 \cdot 7}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

$4 = \frac{2 + 3 \cdot 7}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $3$ by $7$ .

$4 = \frac{2 + 21}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.2.1.5.2

Add $2$ and $21$ .

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - 216 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 3 - 216 a$ with $\frac{1}{3780}$ .

$b = 3 - 216 (\frac{1}{3780})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $3 - 216 (\frac{1}{3780})$ .

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Step 1.3.4.4.1.1

Simplify each term.

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Step 1.3.4.4.1.1.1

Cancel the common factor of $108$ .

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Step 1.3.4.4.1.1.1.1

Factor $108$ out of $- 216$ .

$b = 3 + 108 (- 2) (\frac{1}{3780})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.1.1.2

Factor $108$ out of $3780$ .

$b = 3 + 108 \cdot (- 2 \frac{1}{108 \cdot 35})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.1.1.3

Cancel the common factor.

$b = 3 + 108 \cdot (- 2 \frac{1}{108 \cdot 35})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.1.1.4

Rewrite the expression.

$b = 3 - 2 (\frac{1}{35})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - 2 (\frac{1}{35})$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.1.2

Combine $- 2$ and $\frac{1}{35}$ .

$b = 3 + \frac{- 2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.1.3

Move the negative in front of the fraction.

$b = 3 - \frac{2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = 3 - \frac{2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$b = 3 \cdot \frac{35}{35} - \frac{2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.3

Combine $3$ and $\frac{35}{35}$ .

$b = \frac{3 \cdot 35}{35} - \frac{2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.4

Combine the numerators over the common denominator.

$b = \frac{3 \cdot 35 - 2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.5

Simplify the numerator.

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Step 1.3.4.4.1.5.1

Multiply $3$ by $35$ .

$b = \frac{105 - 2}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.4.4.1.5.2

Subtract $2$ from $105$ .

$b = \frac{103}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = \frac{103}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = \frac{103}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = \frac{103}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

$b = \frac{103}{35}$

$4 = \frac{23}{7}$

$a = \frac{1}{3780}$

Step 1.3.5

Since $4 = \frac{23}{7}$ is not true, there is no solution.

No solution

Step 1.4

Since $y \neq f (x)$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

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Step 2.1

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Step 2.2

Build a set of $3$ equations from the table such that $f (x) = a x^{2} + b x + c$ .

Step 2.3

Calculate the values of $a$ , $b$ , and $c$ .

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Step 2.3.1

Solve for $c$ in $3 = a \cdot 216^{2} + b (216) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 216^{2} + b (216) + c = 3$ .

$a \cdot 216^{2} + b (216) + c = 3$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Raise $216$ to the power of $2$ .

$a \cdot 46656 + b (216) + c = 3$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.1.2.2

Move $46656$ to the left of $a$ .

$46656 \cdot a + b (216) + c = 3$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.1.2.3

Move $216$ to the left of $b$ .

$46656 a + 216 b + c = 3$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

$46656 a + 216 b + c = 3$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.1.3

Move all terms not containing $c$ to the right side of the equation.

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Step 2.3.1.3.1

Subtract $46656 a$ from both sides of the equation.

$216 b + c = 3 - 46656 a$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.1.3.2

Subtract $216 b$ from both sides of the equation.

$c = 3 - 46656 a - 216 b$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

$c = 3 - 46656 a - 216 b$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

$c = 3 - 46656 a - 216 b$

$4 = a \cdot 1296^{2} + b (1296) + c$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2

Replace all occurrences of $c$ with $3 - 46656 a - 216 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $4 = a \cdot 1296^{2} + b (1296) + c$ with $3 - 46656 a - 216 b$ .

$4 = a \cdot 1296^{2} + b (1296) + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2

Simplify $4 = a \cdot 1296^{2} + b (1296) + 3 - 46656 a - 216 b$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$4 = a \cdot 1296^{2} + b (1296) + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = a \cdot 1296^{2} + b (1296) + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 1296^{2} + b (1296) + 3 - 46656 a - 216 b$ .

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Step 2.3.2.2.2.1.1

Simplify each term.

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Step 2.3.2.2.2.1.1.1

Raise $1296$ to the power of $2$ .

$4 = a \cdot 1679616 + b (1296) + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2.2.1.1.2

Move $1679616$ to the left of $a$ .

$4 = 1679616 \cdot a + b (1296) + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2.2.1.1.3

Move $1296$ to the left of $b$ .

$4 = 1679616 a + 1296 b + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = 1679616 a + 1296 b + 3 - 46656 a - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.2.1.2.1

Subtract $46656 a$ from $1679616 a$ .

$4 = 1632960 a + 1296 b + 3 - 216 b$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.2.2.1.2.2

Subtract $216 b$ from $1296 b$ .

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $5 = a \cdot 7776^{2} + b (7776) + c$ with $3 - 46656 a - 216 b$ .

$5 = a \cdot 7776^{2} + b (7776) + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4

Simplify $5 = a \cdot 7776^{2} + b (7776) + 3 - 46656 a - 216 b$ .

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

$5 = a \cdot 7776^{2} + b (7776) + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = a \cdot 7776^{2} + b (7776) + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 7776^{2} + b (7776) + 3 - 46656 a - 216 b$ .

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Step 2.3.2.4.2.1.1

Simplify each term.

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Step 2.3.2.4.2.1.1.1

Raise $7776$ to the power of $2$ .

$5 = a \cdot 60466176 + b (7776) + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4.2.1.1.2

Move $60466176$ to the left of $a$ .

$5 = 60466176 \cdot a + b (7776) + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4.2.1.1.3

Move $7776$ to the left of $b$ .

$5 = 60466176 a + 7776 b + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60466176 a + 7776 b + 3 - 46656 a - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4.2.1.2

Simplify by adding terms.

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Step 2.3.2.4.2.1.2.1

Subtract $46656 a$ from $60466176 a$ .

$5 = 60419520 a + 7776 b + 3 - 216 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.2.4.2.1.2.2

Subtract $216 b$ from $7776 b$ .

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$5 = 60419520 a + 7560 b + 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3

Solve for $a$ in $5 = 60419520 a + 7560 b + 3$ .

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Step 2.3.3.1

Rewrite the equation as $60419520 a + 7560 b + 3 = 5$ .

$60419520 a + 7560 b + 3 = 5$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 2.3.3.2.1

Subtract $7560 b$ from both sides of the equation.

$60419520 a + 3 = 5 - 7560 b$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.2.2

Subtract $3$ from both sides of the equation.

$60419520 a = 5 - 7560 b - 3$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.2.3

Subtract $3$ from $5$ .

$60419520 a = - 7560 b + 2$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$60419520 a = - 7560 b + 2$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3

Divide each term in $60419520 a = - 7560 b + 2$ by $60419520$ and simplify.

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Step 2.3.3.3.1

Divide each term in $60419520 a = - 7560 b + 2$ by $60419520$ .

$\frac{60419520 a}{60419520} = \frac{- 7560 b}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.2

Simplify the left side.

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Step 2.3.3.3.2.1

Cancel the common factor of $60419520$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{60419520 a}{60419520} = \frac{- 7560 b}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 7560 b}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = \frac{- 7560 b}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = \frac{- 7560 b}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3

Simplify the right side.

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Step 2.3.3.3.3.1

Simplify each term.

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Step 2.3.3.3.3.1.1

Cancel the common factor of $- 7560$ and $60419520$ .

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Step 2.3.3.3.3.1.1.1

Factor $7560$ out of $- 7560 b$ .

$a = \frac{7560 (- b)}{60419520} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.3.3.1.1.2.1

Factor $7560$ out of $60419520$ .

$a = \frac{7560 (- b)}{7560 (7992)} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{7560 (- b)}{7560 \cdot 7992} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{7992} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = \frac{- b}{7992} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = \frac{- b}{7992} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{7992} + \frac{2}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.3

Cancel the common factor of $2$ and $60419520$ .

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Step 2.3.3.3.3.1.3.1

Factor $2$ out of $2$ .

$a = - \frac{b}{7992} + \frac{2 (1)}{60419520}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.3.2

Cancel the common factors.

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Step 2.3.3.3.3.1.3.2.1

Factor $2$ out of $60419520$ .

$a = - \frac{b}{7992} + \frac{2 \cdot 1}{2 \cdot 30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{7992} + \frac{2 \cdot 1}{2 \cdot 30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$4 = 1632960 a + 1080 b + 3$

$c = 3 - 46656 a - 216 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{7992} + \frac{1}{30209760}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $4 = 1632960 a + 1080 b + 3$ with $- \frac{b}{7992} + \frac{1}{30209760}$ .

$4 = 1632960 (- \frac{b}{7992} + \frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $1632960 (- \frac{b}{7992} + \frac{1}{30209760}) + 1080 b + 3$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$4 = 1632960 (- \frac{b}{7992}) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.2

Cancel the common factor of $216$ .

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Step 2.3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{7992}$ into the numerator.

$4 = 1632960 (\frac{- b}{7992}) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.2.2

Factor $216$ out of $1632960$ .

$4 = 216 (7560) (\frac{- b}{7992}) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.2.3

Factor $216$ out of $7992$ .

$4 = 216 \cdot (7560 (\frac{- b}{216 \cdot 37})) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.2.4

Cancel the common factor.

$4 = 216 \cdot (7560 (\frac{- b}{216 \cdot 37})) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.2.5

Rewrite the expression.

$4 = 7560 (\frac{- b}{37}) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = 7560 (\frac{- b}{37}) + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.3

Combine $7560$ and $\frac{- b}{37}$ .

$4 = \frac{7560 (- b)}{37} + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.4

Multiply $- 1$ by $7560$ .

$4 = \frac{- 7560 b}{37} + 1632960 (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.5

Cancel the common factor of $816480$ .

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Step 2.3.4.2.1.1.5.1

Factor $816480$ out of $1632960$ .

$4 = \frac{- 7560 b}{37} + 816480 (2) (\frac{1}{30209760}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.5.2

Factor $816480$ out of $30209760$ .

$4 = \frac{- 7560 b}{37} + 816480 \cdot (2 (\frac{1}{816480 \cdot 37})) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.5.3

Cancel the common factor.

$4 = \frac{- 7560 b}{37} + 816480 \cdot (2 (\frac{1}{816480 \cdot 37})) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.5.4

Rewrite the expression.

$4 = \frac{- 7560 b}{37} + 2 (\frac{1}{37}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = \frac{- 7560 b}{37} + 2 (\frac{1}{37}) + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.6

Combine $2$ and $\frac{1}{37}$ .

$4 = \frac{- 7560 b}{37} + \frac{2}{37} + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.1.7

Move the negative in front of the fraction.

$4 = - \frac{7560 b}{37} + \frac{2}{37} + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = - \frac{7560 b}{37} + \frac{2}{37} + 1080 b + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.2

To write $1080 b$ as a fraction with a common denominator, multiply by $\frac{37}{37}$ .

$4 = - \frac{7560 b}{37} + 1080 b \cdot \frac{37}{37} + \frac{2}{37} + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.3

Combine $1080 b$ and $\frac{37}{37}$ .

$4 = - \frac{7560 b}{37} + \frac{1080 b \cdot 37}{37} + \frac{2}{37} + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$4 = \frac{- 7560 b + 1080 b \cdot 37}{37} + \frac{2}{37} + 3$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.5

Combine the numerators over the common denominator.

$4 = 3 + \frac{- 7560 b + 1080 b \cdot 37 + 2}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.6

Multiply $37$ by $1080$ .

$4 = 3 + \frac{- 7560 b + 39960 b + 2}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.7

Add $- 7560 b$ and $39960 b$ .

$4 = 3 + \frac{32400 b + 2}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.8

Factor $2$ out of $32400 b + 2$ .

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Step 2.3.4.2.1.8.1

Factor $2$ out of $32400 b$ .

$4 = 3 + \frac{2 (16200 b) + 2}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.8.2

Factor $2$ out of $2$ .

$4 = 3 + \frac{2 (16200 b) + 2 (1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.8.3

Factor $2$ out of $2 (16200 b) + 2 (1)$ .

$4 = 3 + \frac{2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = 3 + \frac{2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.9

To write $3$ as a fraction with a common denominator, multiply by $\frac{37}{37}$ .

$4 = 3 \cdot \frac{37}{37} + \frac{2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.10

Simplify terms.

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Step 2.3.4.2.1.10.1

Combine $3$ and $\frac{37}{37}$ .

$4 = \frac{3 \cdot 37}{37} + \frac{2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.10.2

Combine the numerators over the common denominator.

$4 = \frac{3 \cdot 37 + 2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = \frac{3 \cdot 37 + 2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.11

Simplify the numerator.

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Step 2.3.4.2.1.11.1

Multiply $3$ by $37$ .

$4 = \frac{111 + 2 (16200 b + 1)}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.11.2

Apply the distributive property.

$4 = \frac{111 + 2 (16200 b) + 2 \cdot 1}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.11.3

Multiply $16200$ by $2$ .

$4 = \frac{111 + 32400 b + 2 \cdot 1}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.11.4

Multiply $2$ by $1$ .

$4 = \frac{111 + 32400 b + 2}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.2.1.11.5

Add $111$ and $2$ .

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 46656 a - 216 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 3 - 46656 a - 216 b$ with $- \frac{b}{7992} + \frac{1}{30209760}$ .

$c = 3 - 46656 (- \frac{b}{7992} + \frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $3 - 46656 (- \frac{b}{7992} + \frac{1}{30209760}) - 216 b$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$c = 3 - 46656 (- \frac{b}{7992}) - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.2

Cancel the common factor of $216$ .

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Step 2.3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{7992}$ into the numerator.

$c = 3 - 46656 \frac{- b}{7992} - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.2.2

Factor $216$ out of $- 46656$ .

$c = 3 + 216 (- 216) (\frac{- b}{7992}) - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.2.3

Factor $216$ out of $7992$ .

$c = 3 + 216 \cdot (- 216 \frac{- b}{216 \cdot 37}) - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.2.4

Cancel the common factor.

$c = 3 + 216 \cdot (- 216 \frac{- b}{216 \cdot 37}) - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.2.5

Rewrite the expression.

$c = 3 - 216 \frac{- b}{37} - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 - 216 \frac{- b}{37} - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.3

Combine $- 216$ and $\frac{- b}{37}$ .

$c = 3 + \frac{- 216 (- b)}{37} - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.4

Multiply $- 1$ by $- 216$ .

$c = 3 + \frac{216 b}{37} - 46656 (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.5

Cancel the common factor of $23328$ .

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Step 2.3.4.4.1.1.5.1

Factor $23328$ out of $- 46656$ .

$c = 3 + \frac{216 b}{37} + 23328 (- 2) (\frac{1}{30209760}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.5.2

Factor $23328$ out of $30209760$ .

$c = 3 + \frac{216 b}{37} + 23328 \cdot (- 2 \frac{1}{23328 \cdot 1295}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.5.3

Cancel the common factor.

$c = 3 + \frac{216 b}{37} + 23328 \cdot (- 2 \frac{1}{23328 \cdot 1295}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.5.4

Rewrite the expression.

$c = 3 + \frac{216 b}{37} - 2 (\frac{1}{1295}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 + \frac{216 b}{37} - 2 (\frac{1}{1295}) - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.6

Combine $- 2$ and $\frac{1}{1295}$ .

$c = 3 + \frac{216 b}{37} + \frac{- 2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.1.7

Move the negative in front of the fraction.

$c = 3 + \frac{216 b}{37} - \frac{2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = 3 + \frac{216 b}{37} - \frac{2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{1295}{1295}$ .

$c = \frac{216 b}{37} + 3 \cdot \frac{1295}{1295} - \frac{2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.3

Combine $3$ and $\frac{1295}{1295}$ .

$c = \frac{216 b}{37} + \frac{3 \cdot 1295}{1295} - \frac{2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{216 b}{37} + \frac{3 \cdot 1295 - 2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $3$ by $1295$ .

$c = \frac{216 b}{37} + \frac{3885 - 2}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.5.2

Subtract $2$ from $3885$ .

$c = \frac{216 b}{37} + \frac{3883}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{216 b}{37} + \frac{3883}{1295} - 216 b$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.6

To write $- 216 b$ as a fraction with a common denominator, multiply by $\frac{37}{37}$ .

$c = \frac{216 b}{37} - 216 b \cdot \frac{37}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.7

Simplify terms.

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Step 2.3.4.4.1.7.1

Combine $- 216 b$ and $\frac{37}{37}$ .

$c = \frac{216 b}{37} + \frac{- 216 b \cdot 37}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.7.2

Combine the numerators over the common denominator.

$c = \frac{216 b - 216 b \cdot 37}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{216 b - 216 b \cdot 37}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8

Simplify each term.

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Step 2.3.4.4.1.8.1

Simplify the numerator.

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Step 2.3.4.4.1.8.1.1

Factor $216 b$ out of $216 b - 216 b \cdot 37$ .

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Step 2.3.4.4.1.8.1.1.1

Factor $216 b$ out of $216 b$ .

$c = \frac{216 b (1) - 216 b \cdot 37}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.1.1.2

Factor $216 b$ out of $- 216 b \cdot 37$ .

$c = \frac{216 b (1) + 216 b (- 1 \cdot 37)}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.1.1.3

Factor $216 b$ out of $216 b (1) + 216 b (- 1 \cdot 37)$ .

$c = \frac{216 b (1 - 1 \cdot 37)}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{216 b (1 - 1 \cdot 37)}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.1.2

Multiply $- 1$ by $37$ .

$c = \frac{216 b (1 - 37)}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.1.3

Subtract $37$ from $1$ .

$c = \frac{216 b \cdot - 36}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.1.4

Multiply $- 36$ by $216$ .

$c = \frac{- 7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{- 7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.4.4.1.8.2

Move the negative in front of the fraction.

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$4 = \frac{32400 b + 113}{37}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5

Solve for $b$ in $4 = \frac{32400 b + 113}{37}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{32400 b + 113}{37} = 4$ .

$\frac{32400 b + 113}{37} = 4$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.2

Multiply both sides by $37$ .

$\frac{32400 b + 113}{37} \cdot 37 = 4 \cdot 37$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.3

Simplify.

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Step 2.3.5.3.1

Simplify the left side.

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Step 2.3.5.3.1.1

Cancel the common factor of $37$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor.

$\frac{32400 b + 113}{37} \cdot 37 = 4 \cdot 37$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.3.1.1.2

Rewrite the expression.

$32400 b + 113 = 4 \cdot 37$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$32400 b + 113 = 4 \cdot 37$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$32400 b + 113 = 4 \cdot 37$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $4$ by $37$ .

$32400 b + 113 = 148$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$32400 b + 113 = 148$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$32400 b + 113 = 148$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4

Solve for $b$ .

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Step 2.3.5.4.1

Move all terms not containing $b$ to the right side of the equation.

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Step 2.3.5.4.1.1

Subtract $113$ from both sides of the equation.

$32400 b = 148 - 113$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.1.2

Subtract $113$ from $148$ .

$32400 b = 35$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$32400 b = 35$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2

Divide each term in $32400 b = 35$ by $32400$ and simplify.

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Step 2.3.5.4.2.1

Divide each term in $32400 b = 35$ by $32400$ .

$\frac{32400 b}{32400} = \frac{35}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.2

Simplify the left side.

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Step 2.3.5.4.2.2.1

Cancel the common factor of $32400$ .

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Step 2.3.5.4.2.2.1.1

Cancel the common factor.

$\frac{32400 b}{32400} = \frac{35}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{35}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{35}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{35}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.3

Simplify the right side.

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Step 2.3.5.4.2.3.1

Cancel the common factor of $35$ and $32400$ .

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Step 2.3.5.4.2.3.1.1

Factor $5$ out of $35$ .

$b = \frac{5 (7)}{32400}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.3.1.2

Cancel the common factors.

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Step 2.3.5.4.2.3.1.2.1

Factor $5$ out of $32400$ .

$b = \frac{5 \cdot 7}{5 \cdot 6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{5 \cdot 7}{5 \cdot 6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$b = \frac{7}{6480}$

$c = - \frac{7776 b}{37} + \frac{3883}{1295}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6

Replace all occurrences of $b$ with $\frac{7}{6480}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - \frac{7776 b}{37} + \frac{3883}{1295}$ with $\frac{7}{6480}$ .

$c = - \frac{7776 (\frac{7}{6480})}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- \frac{7776 (\frac{7}{6480})}{37} + \frac{3883}{1295}$ .

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Step 2.3.6.2.1.1

Simplify each term.

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Step 2.3.6.2.1.1.1

Combine $7776$ and $\frac{7}{6480}$ .

$c = - \frac{\frac{7776 \cdot 7}{6480}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.2

Multiply $7776$ by $7$ .

$c = - \frac{\frac{54432}{6480}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.3

Cancel the common factor of $54432$ and $6480$ .

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Step 2.3.6.2.1.1.3.1

Factor $1296$ out of $54432$ .

$c = - \frac{\frac{1296 (42)}{6480}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.3.2

Cancel the common factors.

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Step 2.3.6.2.1.1.3.2.1

Factor $1296$ out of $6480$ .

$c = - \frac{\frac{1296 \cdot 42}{1296 \cdot 5}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.3.2.2

Cancel the common factor.

$c = - \frac{\frac{1296 \cdot 42}{1296 \cdot 5}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.3.2.3

Rewrite the expression.

$c = - \frac{\frac{42}{5}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{\frac{42}{5}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{\frac{42}{5}}{37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$c = - (\frac{42}{5} \cdot \frac{1}{37}) + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.5

Multiply $\frac{42}{5} \cdot \frac{1}{37}$ .

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Step 2.3.6.2.1.1.5.1

Multiply $\frac{42}{5}$ by $\frac{1}{37}$ .

$c = - \frac{42}{5 \cdot 37} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.1.5.2

Multiply $5$ by $37$ .

$c = - \frac{42}{185} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{42}{185} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{42}{185} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.2

To write $- \frac{42}{185}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$c = - \frac{42}{185} \cdot \frac{7}{7} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.3

Write each expression with a common denominator of $1295$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.2.1.3.1

Multiply $\frac{42}{185}$ by $\frac{7}{7}$ .

$c = - \frac{42 \cdot 7}{185 \cdot 7} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.3.2

Multiply $185$ by $7$ .

$c = - \frac{42 \cdot 7}{1295} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = - \frac{42 \cdot 7}{1295} + \frac{3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.4

Combine the numerators over the common denominator.

$c = \frac{- 42 \cdot 7 + 3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.5

Simplify the numerator.

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Step 2.3.6.2.1.5.1

Multiply $- 42$ by $7$ .

$c = \frac{- 294 + 3883}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.5.2

Add $- 294$ and $3883$ .

$c = \frac{3589}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{3589}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.6

Cancel the common factor of $3589$ and $1295$ .

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Step 2.3.6.2.1.6.1

Factor $37$ out of $3589$ .

$c = \frac{37 (97)}{1295}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.6.2

Cancel the common factors.

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Step 2.3.6.2.1.6.2.1

Factor $37$ out of $1295$ .

$c = \frac{37 \cdot 97}{37 \cdot 35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.6.2.2

Cancel the common factor.

$c = \frac{37 \cdot 97}{37 \cdot 35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.2.1.6.2.3

Rewrite the expression.

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{b}{7992} + \frac{1}{30209760}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{7992} + \frac{1}{30209760}$ with $\frac{7}{6480}$ .

$a = - \frac{\frac{7}{6480}}{7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{\frac{7}{6480}}{7992} + \frac{1}{30209760}$ .

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Step 2.3.6.4.1.1

Simplify each term.

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Step 2.3.6.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (\frac{7}{6480} \cdot \frac{1}{7992}) + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.1.2

Multiply $\frac{7}{6480} \cdot \frac{1}{7992}$ .

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Step 2.3.6.4.1.1.2.1

Multiply $\frac{7}{6480}$ by $\frac{1}{7992}$ .

$a = - \frac{7}{6480 \cdot 7992} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.1.2.2

Multiply $6480$ by $7992$ .

$a = - \frac{7}{51788160} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{7}{51788160} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{7}{51788160} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.2

To write $- \frac{7}{51788160}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$a = - \frac{7}{51788160} \cdot \frac{7}{7} + \frac{1}{30209760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.3

To write $\frac{1}{30209760}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$a = - \frac{7}{51788160} \cdot \frac{7}{7} + \frac{1}{30209760} \cdot \frac{12}{12}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.4

Write each expression with a common denominator of $362517120$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.4.1.4.1

Multiply $\frac{7}{51788160}$ by $\frac{7}{7}$ .

$a = - \frac{7 \cdot 7}{51788160 \cdot 7} + \frac{1}{30209760} \cdot \frac{12}{12}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.4.2

Multiply $51788160$ by $7$ .

$a = - \frac{7 \cdot 7}{362517120} + \frac{1}{30209760} \cdot \frac{12}{12}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.4.3

Multiply $\frac{1}{30209760}$ by $\frac{12}{12}$ .

$a = - \frac{7 \cdot 7}{362517120} + \frac{12}{30209760 \cdot 12}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.4.4

Multiply $30209760$ by $12$ .

$a = - \frac{7 \cdot 7}{362517120} + \frac{12}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{7 \cdot 7}{362517120} + \frac{12}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.5

Combine the numerators over the common denominator.

$a = \frac{- 7 \cdot 7 + 12}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.6

Simplify the numerator.

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Step 2.3.6.4.1.6.1

Multiply $- 7$ by $7$ .

$a = \frac{- 49 + 12}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.6.2

Add $- 49$ and $12$ .

$a = \frac{- 37}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = \frac{- 37}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.7

Cancel the common factor of $- 37$ and $362517120$ .

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Step 2.3.6.4.1.7.1

Factor $37$ out of $- 37$ .

$a = \frac{37 (- 1)}{362517120}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.7.2

Cancel the common factors.

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Step 2.3.6.4.1.7.2.1

Factor $37$ out of $362517120$ .

$a = \frac{37 \cdot - 1}{37 \cdot 9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.7.2.2

Cancel the common factor.

$a = \frac{37 \cdot - 1}{37 \cdot 9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.7.2.3

Rewrite the expression.

$a = \frac{- 1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = \frac{- 1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = \frac{- 1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.6.4.1.8

Move the negative in front of the fraction.

$a = - \frac{1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

$a = - \frac{1}{9797760}$

$c = \frac{97}{35}$

$b = \frac{7}{6480}$

Step 2.3.7

List all of the solutions.

$a = - \frac{1}{9797760}, c = \frac{97}{35}, b = \frac{7}{6480}$

Step 2.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $f (x)$ value in the table.

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Step 2.4.1

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{9797760}$ , $b = \frac{7}{6480}$ , $c = \frac{97}{35}$ , and $x = 216$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise $216$ to the power of $2$ .

$y = - \frac{1}{9797760} \cdot 46656 + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.2

Cancel the common factor of $46656$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{1}{9797760}$ into the numerator.

$y = \frac{- 1}{9797760} \cdot 46656 + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.2.2

Factor $46656$ out of $9797760$ .

$y = \frac{- 1}{46656 (210)} \cdot 46656 + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.2.3

Cancel the common factor.

$y = \frac{- 1}{46656 \cdot 210} \cdot 46656 + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.2.4

Rewrite the expression.

$y = \frac{- 1}{210} + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.3

Move the negative in front of the fraction.

$y = - \frac{1}{210} + (\frac{7}{6480}) \cdot (216) + \frac{97}{35}$

Step 2.4.1.1.4

Cancel the common factor of $216$ .

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Step 2.4.1.1.4.1

Factor $216$ out of $6480$ .

$y = - \frac{1}{210} + \frac{7}{216 (30)} \cdot 216 + \frac{97}{35}$

Step 2.4.1.1.4.2

Cancel the common factor.

$y = - \frac{1}{210} + \frac{7}{216 \cdot 30} \cdot 216 + \frac{97}{35}$

Step 2.4.1.1.4.3

Rewrite the expression.

$y = - \frac{1}{210} + \frac{7}{30} + \frac{97}{35}$

Step 2.4.1.2

Find the common denominator.

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Step 2.4.1.2.1

Multiply $\frac{7}{30}$ by $\frac{7}{7}$ .

$y = - \frac{1}{210} + \frac{7}{30} \cdot \frac{7}{7} + \frac{97}{35}$

Step 2.4.1.2.2

Multiply $\frac{7}{30}$ by $\frac{7}{7}$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{30 \cdot 7} + \frac{97}{35}$

Step 2.4.1.2.3

Multiply $\frac{97}{35}$ by $\frac{6}{6}$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{30 \cdot 7} + \frac{97}{35} \cdot \frac{6}{6}$

Step 2.4.1.2.4

Multiply $\frac{97}{35}$ by $\frac{6}{6}$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{30 \cdot 7} + \frac{97 \cdot 6}{35 \cdot 6}$

Step 2.4.1.2.5

Reorder the factors of $30 \cdot 7$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{7 \cdot 30} + \frac{97 \cdot 6}{35 \cdot 6}$

Step 2.4.1.2.6

Multiply $7$ by $30$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{210} + \frac{97 \cdot 6}{35 \cdot 6}$

Step 2.4.1.2.7

Reorder the factors of $35 \cdot 6$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{210} + \frac{97 \cdot 6}{6 \cdot 35}$

Step 2.4.1.2.8

Multiply $6$ by $35$ .

$y = - \frac{1}{210} + \frac{7 \cdot 7}{210} + \frac{97 \cdot 6}{210}$

Step 2.4.1.3

Combine the numerators over the common denominator.

$y = \frac{- 1 + 7 \cdot 7 + 97 \cdot 6}{210}$

Step 2.4.1.4

Simplify each term.

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Step 2.4.1.4.1

Multiply $7$ by $7$ .

$y = \frac{- 1 + 49 + 97 \cdot 6}{210}$

Step 2.4.1.4.2

Multiply $97$ by $6$ .

$y = \frac{- 1 + 49 + 582}{210}$

Step 2.4.1.5

Simplify the expression.

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Step 2.4.1.5.1

Add $- 1$ and $49$ .

$y = \frac{48 + 582}{210}$

Step 2.4.1.5.2

Add $48$ and $582$ .

$y = \frac{630}{210}$

Step 2.4.1.5.3

Divide $630$ by $210$ .

$y = 3$

Step 2.4.2

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 216$ . This check passes since $y = 3$ and $f (x) = 3$ .

$3 = 3$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{9797760}$ , $b = \frac{7}{6480}$ , $c = \frac{97}{35}$ , and $x = 1296$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $1296$ to the power of $2$ .

$y = - \frac{1}{9797760} \cdot 1679616 + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.2

Cancel the common factor of $279936$ .

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Step 2.4.3.1.2.1

Move the leading negative in $- \frac{1}{9797760}$ into the numerator.

$y = \frac{- 1}{9797760} \cdot 1679616 + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.2.2

Factor $279936$ out of $9797760$ .

$y = \frac{- 1}{279936 (35)} \cdot 1679616 + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.2.3

Factor $279936$ out of $1679616$ .

$y = \frac{- 1}{279936 \cdot 35} \cdot (279936 \cdot 6) + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.2.4

Cancel the common factor.

$y = \frac{- 1}{279936 \cdot 35} \cdot (279936 \cdot 6) + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.2.5

Rewrite the expression.

$y = \frac{- 1}{35} \cdot 6 + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.3

Combine $\frac{- 1}{35}$ and $6$ .

$y = \frac{- 1 \cdot 6}{35} + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.4

Multiply $- 1$ by $6$ .

$y = \frac{- 6}{35} + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.5

Move the negative in front of the fraction.

$y = - \frac{6}{35} + (\frac{7}{6480}) \cdot (1296) + \frac{97}{35}$

Step 2.4.3.1.6

Cancel the common factor of $1296$ .

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Step 2.4.3.1.6.1

Factor $1296$ out of $6480$ .

$y = - \frac{6}{35} + \frac{7}{1296 (5)} \cdot 1296 + \frac{97}{35}$

Step 2.4.3.1.6.2

Cancel the common factor.

$y = - \frac{6}{35} + \frac{7}{1296 \cdot 5} \cdot 1296 + \frac{97}{35}$

Step 2.4.3.1.6.3

Rewrite the expression.

$y = - \frac{6}{35} + \frac{7}{5} + \frac{97}{35}$

Step 2.4.3.2

Simplify terms.

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Step 2.4.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 6 + 97}{35} + \frac{7}{5}$

Step 2.4.3.2.2

Add $- 6$ and $97$ .

$y = \frac{91}{35} + \frac{7}{5}$

Step 2.4.3.2.3

Cancel the common factor of $91$ and $35$ .

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Step 2.4.3.2.3.1

Factor $7$ out of $91$ .

$y = \frac{7 (13)}{35} + \frac{7}{5}$

Step 2.4.3.2.3.2

Cancel the common factors.

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Step 2.4.3.2.3.2.1

Factor $7$ out of $35$ .

$y = \frac{7 \cdot 13}{7 \cdot 5} + \frac{7}{5}$

Step 2.4.3.2.3.2.2

Cancel the common factor.

$y = \frac{7 \cdot 13}{7 \cdot 5} + \frac{7}{5}$

Step 2.4.3.2.3.2.3

Rewrite the expression.

$y = \frac{13}{5} + \frac{7}{5}$

Step 2.4.3.2.4

Combine the numerators over the common denominator.

$y = \frac{13 + 7}{5}$

Step 2.4.3.2.5

Simplify the expression.

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Step 2.4.3.2.5.1

Add $13$ and $7$ .

$y = \frac{20}{5}$

Step 2.4.3.2.5.2

Divide $20$ by $5$ .

$y = 4$

Step 2.4.4

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 1296$ . This check passes since $y = 4$ and $f (x) = 4$ .

$4 = 4$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{9797760}$ , $b = \frac{7}{6480}$ , $c = \frac{97}{35}$ , and $x = 7776$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $7776$ to the power of $2$ .

$y = - \frac{1}{9797760} \cdot 60466176 + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.2

Cancel the common factor of $279936$ .

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Step 2.4.5.1.2.1

Move the leading negative in $- \frac{1}{9797760}$ into the numerator.

$y = \frac{- 1}{9797760} \cdot 60466176 + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.2.2

Factor $279936$ out of $9797760$ .

$y = \frac{- 1}{279936 (35)} \cdot 60466176 + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.2.3

Factor $279936$ out of $60466176$ .

$y = \frac{- 1}{279936 \cdot 35} \cdot (279936 \cdot 216) + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.2.4

Cancel the common factor.

$y = \frac{- 1}{279936 \cdot 35} \cdot (279936 \cdot 216) + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.2.5

Rewrite the expression.

$y = \frac{- 1}{35} \cdot 216 + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.3

Combine $\frac{- 1}{35}$ and $216$ .

$y = \frac{- 1 \cdot 216}{35} + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.4

Multiply $- 1$ by $216$ .

$y = \frac{- 216}{35} + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.5

Move the negative in front of the fraction.

$y = - \frac{216}{35} + (\frac{7}{6480}) \cdot (7776) + \frac{97}{35}$

Step 2.4.5.1.6

Cancel the common factor of $1296$ .

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Step 2.4.5.1.6.1

Factor $1296$ out of $6480$ .

$y = - \frac{216}{35} + \frac{7}{1296 (5)} \cdot 7776 + \frac{97}{35}$

Step 2.4.5.1.6.2

Factor $1296$ out of $7776$ .

$y = - \frac{216}{35} + \frac{7}{1296 \cdot 5} \cdot (1296 \cdot 6) + \frac{97}{35}$

Step 2.4.5.1.6.3

Cancel the common factor.

$y = - \frac{216}{35} + \frac{7}{1296 \cdot 5} \cdot (1296 \cdot 6) + \frac{97}{35}$

Step 2.4.5.1.6.4

Rewrite the expression.

$y = - \frac{216}{35} + \frac{7}{5} \cdot 6 + \frac{97}{35}$

Step 2.4.5.1.7

Combine $\frac{7}{5}$ and $6$ .

$y = - \frac{216}{35} + \frac{7 \cdot 6}{5} + \frac{97}{35}$

Step 2.4.5.1.8

Multiply $7$ by $6$ .

$y = - \frac{216}{35} + \frac{42}{5} + \frac{97}{35}$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = \frac{- 216 + 97}{35} + \frac{42}{5}$

Step 2.4.5.2.2

Add $- 216$ and $97$ .

$y = \frac{- 119}{35} + \frac{42}{5}$

Step 2.4.5.3

Simplify each term.

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Step 2.4.5.3.1

Cancel the common factor of $- 119$ and $35$ .

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Step 2.4.5.3.1.1

Factor $7$ out of $- 119$ .

$y = \frac{7 (- 17)}{35} + \frac{42}{5}$

Step 2.4.5.3.1.2

Cancel the common factors.

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Step 2.4.5.3.1.2.1

Factor $7$ out of $35$ .

$y = \frac{7 \cdot - 17}{7 \cdot 5} + \frac{42}{5}$

Step 2.4.5.3.1.2.2

Cancel the common factor.

$y = \frac{7 \cdot - 17}{7 \cdot 5} + \frac{42}{5}$

Step 2.4.5.3.1.2.3

Rewrite the expression.

$y = \frac{- 17}{5} + \frac{42}{5}$

Step 2.4.5.3.2

Move the negative in front of the fraction.

$y = - \frac{17}{5} + \frac{42}{5}$

Step 2.4.5.4

Combine fractions.

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Step 2.4.5.4.1

Combine the numerators over the common denominator.

$y = \frac{- 17 + 42}{5}$

Step 2.4.5.4.2

Simplify the expression.

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Step 2.4.5.4.2.1

Add $- 17$ and $42$ .

$y = \frac{25}{5}$

Step 2.4.5.4.2.2

Divide $25$ by $5$ .

$y = 5$

Step 2.4.6

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 7776$ . This check passes since $y = 5$ and $f (x) = 5$ .

$5 = 5$

Step 2.4.7

Since $y = f (x)$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = f (x)$ , the function is quadratic and follows the form $y = - \frac{x^{2}}{9797760} + \frac{7 x}{6480} + \frac{97}{35}$ .

$y = - \frac{x^{2}}{9797760} + \frac{7 x}{6480} + \frac{97}{35}$