Algebra Examples

$(3, - 12)$ , $(- 2, 43)$ , $(2, - 5)$

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.

$- 12 = a {(3)}^{2} + b (3) + c, 43 = a {(- 2)}^{2} + b (- 2) + c, - 5 = a {(2)}^{2} + b (2) + c$

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 $c$ in $- 12 = a \cdot 3^{2} + b (3) + c$ .

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Step 3.1.1

Rewrite the equation as $a \cdot 3^{2} + b (3) + c = - 12$ .

$a \cdot 3^{2} + b (3) + c = - 12$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

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

$a \cdot 9 + b (3) + c = - 12$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.1.2.2

Move $9$ to the left of $a$ .

$9 \cdot a + b (3) + c = - 12$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.1.2.3

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

$9 a + 3 b + c = - 12$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

$9 a + 3 b + c = - 12$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.1.3

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

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Step 3.1.3.1

Subtract $9 a$ from both sides of the equation.

$3 b + c = - 12 - 9 a$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.1.3.2

Subtract $3 b$ from both sides of the equation.

$c = - 12 - 9 a - 3 b$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

$c = - 12 - 9 a - 3 b$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

$c = - 12 - 9 a - 3 b$

$43 = a {(- 2)}^{2} + b (- 2) + c$

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

Step 3.2

Replace all occurrences of $c$ with $- 12 - 9 a - 3 b$ in each equation.

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Step 3.2.1

Replace all occurrences of $c$ in $43 = a {(- 2)}^{2} + b (- 2) + c$ with $- 12 - 9 a - 3 b$ .

$43 = a {(- 2)}^{2} + b (- 2) - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2

Simplify $43 = a {(- 2)}^{2} + b (- 2) - 12 - 9 a - 3 b$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$43 = a {(- 2)}^{2} + b (- 2) - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

$43 = a {(- 2)}^{2} + b (- 2) - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $a {(- 2)}^{2} + b (- 2) - 12 - 9 a - 3 b$ .

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Step 3.2.2.2.1.1

Simplify each term.

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Step 3.2.2.2.1.1.1

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

$43 = a \cdot 4 + b (- 2) - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2.2.1.1.2

Move $4$ to the left of $a$ .

$43 = 4 \cdot a + b (- 2) - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2.2.1.1.3

Move $- 2$ to the left of $b$ .

$43 = 4 a - 2 b - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

$43 = 4 a - 2 b - 12 - 9 a - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.2.1.2.1

Subtract $9 a$ from $4 a$ .

$43 = - 5 a - 2 b - 12 - 3 b$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.2.2.1.2.2

Subtract $3 b$ from $- 2 b$ .

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

Step 3.2.3

Replace all occurrences of $c$ in $- 5 = a \cdot 2^{2} + b (2) + c$ with $- 12 - 9 a - 3 b$ .

$- 5 = a \cdot 2^{2} + b (2) - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4

Simplify $- 5 = a \cdot 2^{2} + b (2) - 12 - 9 a - 3 b$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

$- 5 = a \cdot 2^{2} + b (2) - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = a \cdot 2^{2} + b (2) - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify $a \cdot 2^{2} + b (2) - 12 - 9 a - 3 b$ .

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Step 3.2.4.2.1.1

Simplify each term.

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Step 3.2.4.2.1.1.1

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

$- 5 = a \cdot 4 + b (2) - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4.2.1.1.2

Move $4$ to the left of $a$ .

$- 5 = 4 \cdot a + b (2) - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4.2.1.1.3

Move $2$ to the left of $b$ .

$- 5 = 4 a + 2 b - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = 4 a + 2 b - 12 - 9 a - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4.2.1.2

Simplify by adding terms.

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Step 3.2.4.2.1.2.1

Subtract $9 a$ from $4 a$ .

$- 5 = - 5 a + 2 b - 12 - 3 b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.2.4.2.1.2.2

Subtract $3 b$ from $2 b$ .

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 = - 5 a - b - 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3

Solve for $a$ in $- 5 = - 5 a - b - 12$ .

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Step 3.3.1

Rewrite the equation as $- 5 a - b - 12 = - 5$ .

$- 5 a - b - 12 = - 5$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.2

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

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Step 3.3.2.1

Add $b$ to both sides of the equation.

$- 5 a - 12 = - 5 + b$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.2.2

Add $12$ to both sides of the equation.

$- 5 a = - 5 + b + 12$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.2.3

Add $- 5$ and $12$ .

$- 5 a = b + 7$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$- 5 a = b + 7$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.3

Divide each term in $- 5 a = b + 7$ by $- 5$ and simplify.

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Step 3.3.3.1

Divide each term in $- 5 a = b + 7$ by $- 5$ .

$\frac{- 5 a}{- 5} = \frac{b}{- 5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 a}{- 5} = \frac{b}{- 5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{b}{- 5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = \frac{b}{- 5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = \frac{b}{- 5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

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

Move the negative in front of the fraction.

$a = - \frac{b}{5} + \frac{7}{- 5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{5} - \frac{7}{5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = - \frac{b}{5} - \frac{7}{5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = - \frac{b}{5} - \frac{7}{5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = - \frac{b}{5} - \frac{7}{5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

$a = - \frac{b}{5} - \frac{7}{5}$

$43 = - 5 a - 5 b - 12$

$c = - 12 - 9 a - 3 b$

Step 3.4

Replace all occurrences of $a$ with $- \frac{b}{5} - \frac{7}{5}$ in each equation.

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Step 3.4.1

Replace all occurrences of $a$ in $43 = - 5 a - 5 b - 12$ with $- \frac{b}{5} - \frac{7}{5}$ .

$43 = - 5 (- \frac{b}{5} - \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 5 (- \frac{b}{5} - \frac{7}{5}) - 5 b - 12$ .

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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.

$43 = - 5 (- \frac{b}{5}) - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{5}$ into the numerator.

$43 = - 5 \frac{- b}{5} - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.2.2

Factor $5$ out of $- 5$ .

$43 = 5 (- 1) (\frac{- b}{5}) - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.2.3

Cancel the common factor.

$43 = 5 \cdot (- 1 \frac{- b}{5}) - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.2.4

Rewrite the expression.

$43 = - 1 (- b) - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = - 1 (- b) - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.3

Multiply $- 1$ by $- 1$ .

$43 = 1 b - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.4

Multiply $b$ by $1$ .

$43 = b - 5 (- \frac{7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.5

Cancel the common factor of $5$ .

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Step 3.4.2.1.1.5.1

Move the leading negative in $- \frac{7}{5}$ into the numerator.

$43 = b - 5 (\frac{- 7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.5.2

Factor $5$ out of $- 5$ .

$43 = b + 5 (- 1) (\frac{- 7}{5}) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.5.3

Cancel the common factor.

$43 = b + 5 \cdot (- 1 (\frac{- 7}{5})) - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.5.4

Rewrite the expression.

$43 = b - 1 \cdot - 7 - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = b - 1 \cdot - 7 - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.1.6

Multiply $- 1$ by $- 7$ .

$43 = b + 7 - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = b + 7 - 5 b - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Subtract $5 b$ from $b$ .

$43 = - 4 b + 7 - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.2.1.2.2

Subtract $12$ from $7$ .

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 - 9 a - 3 b$

Step 3.4.3

Replace all occurrences of $a$ in $c = - 12 - 9 a - 3 b$ with $- \frac{b}{5} - \frac{7}{5}$ .

$c = - 12 - 9 (- \frac{b}{5} - \frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $- 12 - 9 (- \frac{b}{5} - \frac{7}{5}) - 3 b$ .

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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.

$c = - 12 - 9 (- \frac{b}{5}) - 9 (- \frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.1.2

Multiply $- 9 (- \frac{b}{5})$ .

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Step 3.4.4.1.1.2.1

Multiply $- 1$ by $- 9$ .

$c = - 12 + 9 (\frac{b}{5}) - 9 (- \frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.1.2.2

Combine $9$ and $\frac{b}{5}$ .

$c = - 12 + \frac{9 b}{5} - 9 (- \frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 + \frac{9 b}{5} - 9 (- \frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.1.3

Multiply $- 9 (- \frac{7}{5})$ .

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Step 3.4.4.1.1.3.1

Multiply $- 1$ by $- 9$ .

$c = - 12 + \frac{9 b}{5} + 9 (\frac{7}{5}) - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.1.3.2

Combine $9$ and $\frac{7}{5}$ .

$c = - 12 + \frac{9 b}{5} + \frac{9 \cdot 7}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.1.3.3

Multiply $9$ by $7$ .

$c = - 12 + \frac{9 b}{5} + \frac{63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 + \frac{9 b}{5} + \frac{63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - 12 + \frac{9 b}{5} + \frac{63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$c = \frac{9 b}{5} - 12 \cdot \frac{5}{5} + \frac{63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.3

Combine $- 12$ and $\frac{5}{5}$ .

$c = \frac{9 b}{5} + \frac{- 12 \cdot 5}{5} + \frac{63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{9 b}{5} + \frac{- 12 \cdot 5 + 63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $- 12$ by $5$ .

$c = \frac{9 b}{5} + \frac{- 60 + 63}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.5.2

Add $- 60$ and $63$ .

$c = \frac{9 b}{5} + \frac{3}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = \frac{9 b}{5} + \frac{3}{5} - 3 b$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.6

To write $- 3 b$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$c = \frac{9 b}{5} - 3 b \cdot \frac{5}{5} + \frac{3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.7

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

$c = \frac{9 b}{5} + \frac{- 3 b \cdot 5}{5} + \frac{3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.8

Combine the numerators over the common denominator.

$c = \frac{9 b - 3 b \cdot 5}{5} + \frac{3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.9

Combine the numerators over the common denominator.

$c = \frac{9 b - 3 b \cdot 5 + 3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.10

Multiply $5$ by $- 3$ .

$c = \frac{9 b - 15 b + 3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.11

Subtract $15 b$ from $9 b$ .

$c = \frac{- 6 b + 3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.12

Factor $3$ out of $- 6 b + 3$ .

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Step 3.4.4.1.12.1

Factor $3$ out of $- 6 b$ .

$c = \frac{3 (- 2 b) + 3}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.12.2

Factor $3$ out of $3$ .

$c = \frac{3 (- 2 b) + 3 (1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.12.3

Factor $3$ out of $3 (- 2 b) + 3 (1)$ .

$c = \frac{3 (- 2 b + 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = \frac{3 (- 2 b + 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.13

Factor $- 1$ out of $- 2 b$ .

$c = \frac{3 (- (2 b) + 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.14

Rewrite $1$ as $- 1 (- 1)$ .

$c = \frac{3 (- (2 b) - 1 \cdot - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.15

Factor $- 1$ out of $- (2 b) - 1 (- 1)$ .

$c = \frac{3 (- (2 b - 1))}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.16

Simplify the expression.

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Step 3.4.4.1.16.1

Rewrite $- (2 b - 1)$ as $- 1 (2 b - 1)$ .

$c = \frac{3 (- 1 (2 b - 1))}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.4.4.1.16.2

Move the negative in front of the fraction.

$c = - \frac{3 (2 b - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - \frac{3 (2 b - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - \frac{3 (2 b - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - \frac{3 (2 b - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - \frac{3 (2 b - 1)}{5}$

$43 = - 4 b - 5$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5

Solve for $b$ in $43 = - 4 b - 5$ .

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Step 3.5.1

Rewrite the equation as $- 4 b - 5 = 43$ .

$- 4 b - 5 = 43$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.2

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

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Step 3.5.2.1

Add $5$ to both sides of the equation.

$- 4 b = 43 + 5$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.2.2

Add $43$ and $5$ .

$- 4 b = 48$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$- 4 b = 48$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.3

Divide each term in $- 4 b = 48$ by $- 4$ and simplify.

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Step 3.5.3.1

Divide each term in $- 4 b = 48$ by $- 4$ .

$\frac{- 4 b}{- 4} = \frac{48}{- 4}$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $- 4$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{- 4 b}{- 4} = \frac{48}{- 4}$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{48}{- 4}$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$b = \frac{48}{- 4}$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$b = \frac{48}{- 4}$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $48$ by $- 4$ .

$b = - 12$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$b = - 12$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$b = - 12$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

$b = - 12$

$c = - \frac{3 (2 b - 1)}{5}$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6

Replace all occurrences of $b$ with $- 12$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $c = - \frac{3 (2 b - 1)}{5}$ with $- 12$ .

$c = - \frac{3 (2 (- 12) - 1)}{5}$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{3 (2 (- 12) - 1)}{5}$ .

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Step 3.6.2.1.1

Simplify the numerator.

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Step 3.6.2.1.1.1

Multiply $2$ by $- 12$ .

$c = - \frac{3 (- 24 - 1)}{5}$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.2.1.1.2

Subtract $1$ from $- 24$ .

$c = - \frac{3 \cdot - 25}{5}$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = - \frac{3 \cdot - 25}{5}$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.2.1.2

Simplify the expression.

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Step 3.6.2.1.2.1

Multiply $3$ by $- 25$ .

$c = - \frac{- 75}{5}$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.2.1.2.2

Divide $- 75$ by $5$ .

$c = 15$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.2.1.2.3

Multiply $- 1$ by $- 15$ .

$c = 15$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = 15$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = 15$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

$c = 15$

$b = - 12$

$a = - \frac{b}{5} - \frac{7}{5}$

Step 3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{5} - \frac{7}{5}$ with $- 12$ .

$a = - \frac{- 12}{5} - \frac{7}{5}$

$c = 15$

$b = - 12$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- \frac{- 12}{5} - \frac{7}{5}$ .

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Step 3.6.4.1.1

Combine the numerators over the common denominator.

$a = \frac{12 - 7}{5}$

$c = 15$

$b = - 12$

Step 3.6.4.1.2

Simplify the expression.

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Step 3.6.4.1.2.1

Subtract $7$ from $12$ .

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

$c = 15$

$b = - 12$

Step 3.6.4.1.2.2

Divide $5$ by $5$ .

$a = 1$

$c = 15$

$b = - 12$

$a = 1$

$c = 15$

$b = - 12$

$a = 1$

$c = 15$

$b = - 12$

$a = 1$

$c = 15$

$b = - 12$

$a = 1$

$c = 15$

$b = - 12$

Step 3.7

List all of the solutions.

$a = 1, c = 15, b = - 12$

Step 4

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

$y = x^{2} - 12 x + 15$

Step 5