Algebra Examples

$(1, 14)$ , $(3, 10)$ , $(5, 16)$

Step 1

Use the standard form of a quadratic equation $y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

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

Step 2

Create a system of equations by substituting the $x$ and $y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

$14 = a {(1)}^{2} + b (1) + c, 10 = a {(3)}^{2} + b (3) + c, 16 = a {(5)}^{2} + b (5) + c$

Step 3

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

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Step 3.1

Solve for $a$ in $14 = a + b + c$ .

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Step 3.1.1

Rewrite the equation as $a + b + c = 14$ .

$a + b + c = 14$

$10 = a \cdot 3^{2} + b (3) + c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.1.2

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

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Step 3.1.2.1

Subtract $b$ from both sides of the equation.

$a + c = 14 - b$

$10 = a \cdot 3^{2} + b (3) + c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.1.2.2

Subtract $c$ from both sides of the equation.

$a = 14 - b - c$

$10 = a \cdot 3^{2} + b (3) + c$

$16 = a \cdot 5^{2} + b (5) + c$

$a = 14 - b - c$

$10 = a \cdot 3^{2} + b (3) + c$

$16 = a \cdot 5^{2} + b (5) + c$

$a = 14 - b - c$

$10 = a \cdot 3^{2} + b (3) + c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2

Replace all occurrences of $a$ with $14 - b - c$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $10 = a \cdot 3^{2} + b (3) + c$ with $14 - b - c$ .

$10 = (14 - b - c) \cdot 3^{2} + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(14 - b - c) \cdot 3^{2} + b (3) + c$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Raise $3$ to the power of $2$ .

$10 = (14 - b - c) \cdot 9 + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.1.2

Apply the distributive property.

$10 = 14 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.1.3

Simplify.

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Step 3.2.2.1.1.3.1

Multiply $14$ by $9$ .

$10 = 126 - b \cdot 9 - c \cdot 9 + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.1.3.2

Multiply $9$ by $- 1$ .

$10 = 126 - 9 b - c \cdot 9 + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.1.3.3

Multiply $9$ by $- 1$ .

$10 = 126 - 9 b - 9 c + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

$10 = 126 - 9 b - 9 c + b (3) + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.1.4

Move $3$ to the left of $b$ .

$10 = 126 - 9 b - 9 c + 3 b + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

$10 = 126 - 9 b - 9 c + 3 b + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.1.2.1

Add $- 9 b$ and $3 b$ .

$10 = 126 - 6 b - 9 c + c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.1.2.2

Add $- 9 c$ and $c$ .

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = a \cdot 5^{2} + b (5) + c$

Step 3.2.3

Replace all occurrences of $a$ in $16 = a \cdot 5^{2} + b (5) + c$ with $14 - b - c$ .

$16 = (14 - b - c) \cdot 5^{2} + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $(14 - b - c) \cdot 5^{2} + b (5) + c$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Raise $5$ to the power of $2$ .

$16 = (14 - b - c) \cdot 25 + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.1.2

Apply the distributive property.

$16 = 14 \cdot 25 - b \cdot 25 - c \cdot 25 + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.1.3

Simplify.

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Step 3.2.4.1.1.3.1

Multiply $14$ by $25$ .

$16 = 350 - b \cdot 25 - c \cdot 25 + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.1.3.2

Multiply $25$ by $- 1$ .

$16 = 350 - 25 b - c \cdot 25 + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.1.3.3

Multiply $25$ by $- 1$ .

$16 = 350 - 25 b - 25 c + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 25 b - 25 c + b (5) + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.1.4

Move $5$ to the left of $b$ .

$16 = 350 - 25 b - 25 c + 5 b + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 25 b - 25 c + 5 b + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.2

Simplify by adding terms.

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Step 3.2.4.1.2.1

Add $- 25 b$ and $5 b$ .

$16 = 350 - 20 b - 25 c + c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.2.4.1.2.2

Add $- 25 c$ and $c$ .

$16 = 350 - 20 b - 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 20 b - 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 20 b - 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 20 b - 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$16 = 350 - 20 b - 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3

Solve for $b$ in $16 = 350 - 20 b - 24 c$ .

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Step 3.3.1

Rewrite the equation as $350 - 20 b - 24 c = 16$ .

$350 - 20 b - 24 c = 16$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $350$ from both sides of the equation.

$- 20 b - 24 c = 16 - 350$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.2.2

Add $24 c$ to both sides of the equation.

$- 20 b = 16 - 350 + 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.2.3

Subtract $350$ from $16$ .

$- 20 b = - 334 + 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$- 20 b = - 334 + 24 c$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3

Divide each term in $- 20 b = - 334 + 24 c$ by $- 20$ and simplify.

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Step 3.3.3.1

Divide each term in $- 20 b = - 334 + 24 c$ by $- 20$ .

$\frac{- 20 b}{- 20} = \frac{- 334}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 20$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 20 b}{- 20} = \frac{- 334}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 334}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{- 334}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{- 334}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Cancel the common factor of $- 334$ and $- 20$ .

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Step 3.3.3.3.1.1.1

Factor $- 2$ out of $- 334$ .

$b = \frac{- 2 \cdot 167}{- 20} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.3.1.1.2.1

Factor $- 2$ out of $- 20$ .

$b = \frac{- 2 \cdot 167}{- 2 \cdot 10} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$b = \frac{- 2 \cdot 167}{- 2 \cdot 10} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$b = \frac{167}{10} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} + \frac{24 c}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.2

Cancel the common factor of $24$ and $- 20$ .

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Step 3.3.3.3.1.2.1

Factor $4$ out of $24 c$ .

$b = \frac{167}{10} + \frac{4 (6 c)}{- 20}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.3.1.2.2.1

Factor $4$ out of $- 20$ .

$b = \frac{167}{10} + \frac{4 (6 c)}{4 (- 5)}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.2.2.2

Cancel the common factor.

$b = \frac{167}{10} + \frac{4 (6 c)}{4 \cdot - 5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.2.2.3

Rewrite the expression.

$b = \frac{167}{10} + \frac{6 c}{- 5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} + \frac{6 c}{- 5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} + \frac{6 c}{- 5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.3.3.3.1.3

Move the negative in front of the fraction.

$b = \frac{167}{10} - \frac{6 c}{5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$10 = 126 - 6 b - 8 c$

$a = 14 - b - c$

Step 3.4

Replace all occurrences of $b$ with $\frac{167}{10} - \frac{6 c}{5}$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $10 = 126 - 6 b - 8 c$ with $\frac{167}{10} - \frac{6 c}{5}$ .

$10 = 126 - 6 (\frac{167}{10} - \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $126 - 6 (\frac{167}{10} - \frac{6 c}{5}) - 8 c$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$10 = 126 - 6 (\frac{167}{10}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Factor $2$ out of $- 6$ .

$10 = 126 + 2 (- 3) (\frac{167}{10}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.2.2

Factor $2$ out of $10$ .

$10 = 126 + 2 \cdot (- 3 \frac{167}{2 \cdot 5}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.2.3

Cancel the common factor.

$10 = 126 + 2 \cdot (- 3 \frac{167}{2 \cdot 5}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.2.4

Rewrite the expression.

$10 = 126 - 3 (\frac{167}{5}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = 126 - 3 (\frac{167}{5}) - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.3

Combine $- 3$ and $\frac{167}{5}$ .

$10 = 126 + \frac{- 3 \cdot 167}{5} - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.4

Multiply $- 3$ by $167$ .

$10 = 126 + \frac{- 501}{5} - 6 (- \frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.5

Multiply $- 6 (- \frac{6 c}{5})$ .

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Step 3.4.2.1.1.5.1

Multiply $- 1$ by $- 6$ .

$10 = 126 + \frac{- 501}{5} + 6 (\frac{6 c}{5}) - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.5.2

Combine $6$ and $\frac{6 c}{5}$ .

$10 = 126 + \frac{- 501}{5} + \frac{6 (6 c)}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.5.3

Multiply $6$ by $6$ .

$10 = 126 + \frac{- 501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = 126 + \frac{- 501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.1.6

Move the negative in front of the fraction.

$10 = 126 - \frac{501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = 126 - \frac{501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.2

To write $126$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$10 = 126 \cdot \frac{5}{5} - \frac{501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.3

Combine $126$ and $\frac{5}{5}$ .

$10 = \frac{126 \cdot 5}{5} - \frac{501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$10 = \frac{126 \cdot 5 - 501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.5

Simplify the numerator.

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Step 3.4.2.1.5.1

Multiply $126$ by $5$ .

$10 = \frac{630 - 501}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.5.2

Subtract $501$ from $630$ .

$10 = \frac{129}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = \frac{129}{5} + \frac{36 c}{5} - 8 c$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.6

To write $- 8 c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$10 = \frac{129}{5} + \frac{36 c}{5} - 8 c \cdot \frac{5}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.7

Combine $- 8 c$ and $\frac{5}{5}$ .

$10 = \frac{129}{5} + \frac{36 c}{5} + \frac{- 8 c \cdot 5}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.8

Combine the numerators over the common denominator.

$10 = \frac{129}{5} + \frac{36 c - 8 c \cdot 5}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.9

Combine the numerators over the common denominator.

$10 = \frac{129 + 36 c - 8 c \cdot 5}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.10

Multiply $5$ by $- 8$ .

$10 = \frac{129 + 36 c - 40 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.2.1.11

Subtract $40 c$ from $36 c$ .

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - b - c$

Step 3.4.3

Replace all occurrences of $b$ in $a = 14 - b - c$ with $\frac{167}{10} - \frac{6 c}{5}$ .

$a = 14 - (\frac{167}{10} - \frac{6 c}{5}) - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $14 - (\frac{167}{10} - \frac{6 c}{5}) - c$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$a = 14 - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.1.2

Multiply $- - \frac{6 c}{5}$ .

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Step 3.4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$a = 14 - \frac{167}{10} + 1 (\frac{6 c}{5}) - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.1.2.2

Multiply $\frac{6 c}{5}$ by $1$ .

$a = 14 - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = 14 - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.2

To write $14$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$a = 14 \cdot \frac{10}{10} - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.3

Combine $14$ and $\frac{10}{10}$ .

$a = \frac{14 \cdot 10}{10} - \frac{167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$a = \frac{14 \cdot 10 - 167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $14$ by $10$ .

$a = \frac{140 - 167}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.5.2

Subtract $167$ from $140$ .

$a = \frac{- 27}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{- 27}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.6

Move the negative in front of the fraction.

$a = - \frac{27}{10} + \frac{6 c}{5} - c$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.7

To write $- c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a = - \frac{27}{10} + \frac{6 c}{5} - c \cdot \frac{5}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.8

Simplify terms.

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Step 3.4.4.1.8.1

Combine $- c$ and $\frac{5}{5}$ .

$a = - \frac{27}{10} + \frac{6 c}{5} + \frac{- c \cdot 5}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.8.2

Combine the numerators over the common denominator.

$a = - \frac{27}{10} + \frac{6 c - c \cdot 5}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{6 c - c \cdot 5}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9

Simplify each term.

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Step 3.4.4.1.9.1

Simplify the numerator.

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Step 3.4.4.1.9.1.1

Factor $c$ out of $6 c - c \cdot 5$ .

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Step 3.4.4.1.9.1.1.1

Factor $c$ out of $6 c$ .

$a = - \frac{27}{10} + \frac{c \cdot 6 - c \cdot 5}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9.1.1.2

Factor $c$ out of $- c \cdot 5$ .

$a = - \frac{27}{10} + \frac{c \cdot 6 + c (- 1 \cdot 5)}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9.1.1.3

Factor $c$ out of $c \cdot 6 + c (- 1 \cdot 5)$ .

$a = - \frac{27}{10} + \frac{c (6 - 1 \cdot 5)}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c (6 - 1 \cdot 5)}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9.1.2

Multiply $- 1$ by $5$ .

$a = - \frac{27}{10} + \frac{c (6 - 5)}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9.1.3

Subtract $5$ from $6$ .

$a = - \frac{27}{10} + \frac{c \cdot 1}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c \cdot 1}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.4.4.1.9.2

Multiply $c$ by $1$ .

$a = - \frac{27}{10} + \frac{c}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{c}{5}$

$10 = \frac{129 - 4 c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5

Solve for $c$ in $10 = \frac{129 - 4 c}{5}$ .

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Step 3.5.1

Rewrite the equation as $\frac{129 - 4 c}{5} = 10$ .

$\frac{129 - 4 c}{5} = 10$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.2

Multiply both sides by $5$ .

$\frac{129 - 4 c}{5} \cdot 5 = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Simplify $\frac{129 - 4 c}{5} \cdot 5$ .

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Step 3.5.3.1.1.1

Cancel the common factor of $5$ .

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Step 3.5.3.1.1.1.1

Cancel the common factor.

$\frac{129 - 4 c}{5} \cdot 5 = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.3.1.1.1.2

Rewrite the expression.

$129 - 4 c = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$129 - 4 c = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.3.1.1.2

Reorder $129$ and $- 4 c$ .

$- 4 c + 129 = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$- 4 c + 129 = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$- 4 c + 129 = 10 \cdot 5$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Multiply $10$ by $5$ .

$- 4 c + 129 = 50$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$- 4 c + 129 = 50$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$- 4 c + 129 = 50$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4

Solve for $c$ .

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Step 3.5.4.1

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

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Step 3.5.4.1.1

Subtract $129$ from both sides of the equation.

$- 4 c = 50 - 129$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4.1.2

Subtract $129$ from $50$ .

$- 4 c = - 79$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$- 4 c = - 79$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4.2

Divide each term in $- 4 c = - 79$ by $- 4$ and simplify.

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Step 3.5.4.2.1

Divide each term in $- 4 c = - 79$ by $- 4$ .

$\frac{- 4 c}{- 4} = \frac{- 79}{- 4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4.2.2

Simplify the left side.

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Step 3.5.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.5.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 c}{- 4} = \frac{- 79}{- 4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 79}{- 4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{- 79}{- 4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{- 79}{- 4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.5.4.2.3

Simplify the right side.

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Step 3.5.4.2.3.1

Dividing two negative values results in a positive value.

$c = \frac{79}{4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{79}{4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{79}{4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{79}{4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$c = \frac{79}{4}$

$a = - \frac{27}{10} + \frac{c}{5}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6

Replace all occurrences of $c$ with $\frac{79}{4}$ in each equation.

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Step 3.6.1

Replace all occurrences of $c$ in $a = - \frac{27}{10} + \frac{c}{5}$ with $\frac{79}{4}$ .

$a = - \frac{27}{10} + \frac{\frac{79}{4}}{5}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{27}{10} + \frac{\frac{79}{4}}{5}$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - \frac{27}{10} + \frac{79}{4} \cdot \frac{1}{5}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.1.2

Multiply $\frac{79}{4} \cdot \frac{1}{5}$ .

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Step 3.6.2.1.1.2.1

Multiply $\frac{79}{4}$ by $\frac{1}{5}$ .

$a = - \frac{27}{10} + \frac{79}{4 \cdot 5}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.1.2.2

Multiply $4$ by $5$ .

$a = - \frac{27}{10} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27}{10} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.2

To write $- \frac{27}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = - \frac{27}{10} \cdot \frac{2}{2} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.2.1.3.1

Multiply $\frac{27}{10}$ by $\frac{2}{2}$ .

$a = - \frac{27 \cdot 2}{10 \cdot 2} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.3.2

Multiply $10$ by $2$ .

$a = - \frac{27 \cdot 2}{20} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = - \frac{27 \cdot 2}{20} + \frac{79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.4

Combine the numerators over the common denominator.

$a = \frac{- 27 \cdot 2 + 79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.5

Simplify the numerator.

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Step 3.6.2.1.5.1

Multiply $- 27$ by $2$ .

$a = \frac{- 54 + 79}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.5.2

Add $- 54$ and $79$ .

$a = \frac{25}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{25}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.6

Cancel the common factor of $25$ and $20$ .

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Step 3.6.2.1.6.1

Factor $5$ out of $25$ .

$a = \frac{5 (5)}{20}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.6.2

Cancel the common factors.

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Step 3.6.2.1.6.2.1

Factor $5$ out of $20$ .

$a = \frac{5 \cdot 5}{5 \cdot 4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.6.2.2

Cancel the common factor.

$a = \frac{5 \cdot 5}{5 \cdot 4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.2.1.6.2.3

Rewrite the expression.

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{6 c}{5}$

Step 3.6.3

Replace all occurrences of $c$ in $b = \frac{167}{10} - \frac{6 c}{5}$ with $\frac{79}{4}$ .

$b = \frac{167}{10} - \frac{6 (\frac{79}{4})}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $\frac{167}{10} - \frac{6 (\frac{79}{4})}{5}$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Combine $6$ and $\frac{79}{4}$ .

$b = \frac{167}{10} - \frac{\frac{6 \cdot 79}{4}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.2

Multiply $6$ by $79$ .

$b = \frac{167}{10} - \frac{\frac{474}{4}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.3

Cancel the common factor of $474$ and $4$ .

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Step 3.6.4.1.1.3.1

Factor $2$ out of $474$ .

$b = \frac{167}{10} - \frac{\frac{2 (237)}{4}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.3.2

Cancel the common factors.

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Step 3.6.4.1.1.3.2.1

Factor $2$ out of $4$ .

$b = \frac{167}{10} - \frac{\frac{2 \cdot 237}{2 \cdot 2}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.3.2.2

Cancel the common factor.

$b = \frac{167}{10} - \frac{\frac{2 \cdot 237}{2 \cdot 2}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.3.2.3

Rewrite the expression.

$b = \frac{167}{10} - \frac{\frac{237}{2}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{\frac{237}{2}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{\frac{237}{2}}{5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$b = \frac{167}{10} - (\frac{237}{2} \cdot \frac{1}{5})$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.5

Multiply $\frac{237}{2} \cdot \frac{1}{5}$ .

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Step 3.6.4.1.1.5.1

Multiply $\frac{237}{2}$ by $\frac{1}{5}$ .

$b = \frac{167}{10} - \frac{237}{2 \cdot 5}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.1.5.2

Multiply $2$ by $5$ .

$b = \frac{167}{10} - \frac{237}{10}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{237}{10}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = \frac{167}{10} - \frac{237}{10}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.2

Combine fractions.

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Step 3.6.4.1.2.1

Combine the numerators over the common denominator.

$b = \frac{167 - 237}{10}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.2.2

Simplify the expression.

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Step 3.6.4.1.2.2.1

Subtract $237$ from $167$ .

$b = \frac{- 70}{10}$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.6.4.1.2.2.2

Divide $- 70$ by $10$ .

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

$b = - 7$

$a = \frac{5}{4}$

$c = \frac{79}{4}$

Step 3.7

List all of the solutions.

$b = - 7, a = \frac{5}{4}, c = \frac{79}{4}$

Step 4

Substitute the actual values of $a$ , $b$ , and $c$ into the formula for a quadratic equation to find the resulting equation.

$y = \frac{5 x^{2}}{4} - 7 x + \frac{79}{4}$

Step 5