Finite Math Examples

$(2, 0)$ , $(3, - 2)$ , $(1, - 2)$

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.

$0 = a {(2)}^{2} + b (2) + c, - 2 = a {(3)}^{2} + b (3) + c, - 2 = a {(1)}^{2} + b (1) + c$

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 $- 2 = a + b + c$ .

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Step 3.1.1

Rewrite the equation as $a + b + c = - 2$ .

$a + b + c = - 2$

$0 = a \cdot 2^{2} + b (2) + c$

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

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.

$a + c = - 2 - b$

$0 = a \cdot 2^{2} + b (2) + c$

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

Step 3.1.2.2

Subtract $c$ from both sides of the equation.

$a = - 2 - b - c$

$0 = a \cdot 2^{2} + b (2) + c$

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

$a = - 2 - b - c$

$0 = a \cdot 2^{2} + b (2) + c$

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

$a = - 2 - b - c$

$0 = a \cdot 2^{2} + b (2) + c$

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

Step 3.2

Replace all occurrences of $a$ with $- 2 - b - c$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $0 = a \cdot 2^{2} + b (2) + c$ with $- 2 - b - c$ .

$0 = (- 2 - b - c) \cdot 2^{2} + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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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 $2$ to the power of $2$ .

$0 = (- 2 - b - c) \cdot 4 + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.1.2

Apply the distributive property.

$0 = - 2 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.1.3

Simplify.

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Step 3.2.2.1.1.3.1

Multiply $- 2$ by $4$ .

$0 = - 8 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.1.3.2

Multiply $4$ by $- 1$ .

$0 = - 8 - 4 b - c \cdot 4 + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.1.3.3

Multiply $4$ by $- 1$ .

$0 = - 8 - 4 b - 4 c + b (2) + c$

$a = - 2 - b - c$

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

$0 = - 8 - 4 b - 4 c + b (2) + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.1.4

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

$0 = - 8 - 4 b - 4 c + 2 b + c$

$a = - 2 - b - c$

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

$0 = - 8 - 4 b - 4 c + 2 b + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.1.2.1

Add $- 4 b$ and $2 b$ .

$0 = - 8 - 2 b - 4 c + c$

$a = - 2 - b - c$

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

Step 3.2.2.1.2.2

Add $- 4 c$ and $c$ .

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

Step 3.2.3

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

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

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

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

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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 $3$ to the power of $2$ .

$- 2 = (- 2 - b - c) \cdot 9 + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.1.2

Apply the distributive property.

$- 2 = - 2 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.1.3

Simplify.

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Step 3.2.4.1.1.3.1

Multiply $- 2$ by $9$ .

$- 2 = - 18 - b \cdot 9 - c \cdot 9 + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.1.3.2

Multiply $9$ by $- 1$ .

$- 2 = - 18 - 9 b - c \cdot 9 + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.1.3.3

Multiply $9$ by $- 1$ .

$- 2 = - 18 - 9 b - 9 c + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 9 b - 9 c + b (3) + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.1.4

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

$- 2 = - 18 - 9 b - 9 c + 3 b + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 9 b - 9 c + 3 b + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.2

Simplify by adding terms.

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Step 3.2.4.1.2.1

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

$- 2 = - 18 - 6 b - 9 c + c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.2.4.1.2.2

Add $- 9 c$ and $c$ .

$- 2 = - 18 - 6 b - 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 6 b - 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 6 b - 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 6 b - 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 2 = - 18 - 6 b - 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3

Solve for $b$ in $- 2 = - 18 - 6 b - 8 c$ .

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Step 3.3.1

Rewrite the equation as $- 18 - 6 b - 8 c = - 2$ .

$- 18 - 6 b - 8 c = - 2$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

Add $18$ to both sides of the equation.

$- 6 b - 8 c = - 2 + 18$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.2.2

Add $8 c$ to both sides of the equation.

$- 6 b = - 2 + 18 + 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.2.3

Add $- 2$ and $18$ .

$- 6 b = 16 + 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 6 b = 16 + 8 c$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3

Divide each term in $- 6 b = 16 + 8 c$ by $- 6$ and simplify.

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Step 3.3.3.1

Divide each term in $- 6 b = 16 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{16}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{16}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{16}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{16}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{16}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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 $16$ and $- 6$ .

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Step 3.3.3.3.1.1.1

Factor $2$ out of $16$ .

$b = \frac{2 (8)}{- 6} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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 $- 6$ .

$b = \frac{2 \cdot 8}{2 \cdot - 3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$b = \frac{2 \cdot 8}{2 \cdot - 3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$b = \frac{8}{- 3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{8}{- 3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{8}{- 3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$b = - \frac{8}{3} + \frac{8 c}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.3

Cancel the common factor of $8$ and $- 6$ .

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Step 3.3.3.3.1.3.1

Factor $2$ out of $8 c$ .

$b = - \frac{8}{3} + \frac{2 (4 c)}{- 6}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $2$ out of $- 6$ .

$b = - \frac{8}{3} + \frac{2 (4 c)}{2 (- 3)}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$b = - \frac{8}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$b = - \frac{8}{3} + \frac{4 c}{- 3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} + \frac{4 c}{- 3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} + \frac{4 c}{- 3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.3.3.3.1.4

Move the negative in front of the fraction.

$b = - \frac{8}{3} - \frac{4 c}{3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$0 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Step 3.4

Replace all occurrences of $b$ with $- \frac{8}{3} - \frac{4 c}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $0 = - 8 - 2 b - 3 c$ with $- \frac{8}{3} - \frac{4 c}{3}$ .

$0 = - 8 - 2 (- \frac{8}{3} - \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 8 - 2 (- \frac{8}{3} - \frac{4 c}{3}) - 3 c$ .

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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.

$0 = - 8 - 2 (- \frac{8}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.2

Multiply $- 2 (- \frac{8}{3})$ .

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Step 3.4.2.1.1.2.1

Multiply $- 1$ by $- 2$ .

$0 = - 8 + 2 (\frac{8}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.2.2

Combine $2$ and $\frac{8}{3}$ .

$0 = - 8 + \frac{2 \cdot 8}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.2.3

Multiply $2$ by $8$ .

$0 = - 8 + \frac{16}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = - 8 + \frac{16}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.3

Multiply $- 2 (- \frac{4 c}{3})$ .

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Step 3.4.2.1.1.3.1

Multiply $- 1$ by $- 2$ .

$0 = - 8 + \frac{16}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.3.2

Combine $2$ and $\frac{4 c}{3}$ .

$0 = - 8 + \frac{16}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.1.3.3

Multiply $4$ by $2$ .

$0 = - 8 + \frac{16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = - 8 + \frac{16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = - 8 + \frac{16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$0 = - 8 \cdot \frac{3}{3} + \frac{16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.3

Combine $- 8$ and $\frac{3}{3}$ .

$0 = \frac{- 8 \cdot 3}{3} + \frac{16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$0 = \frac{- 8 \cdot 3 + 16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.5

Simplify the numerator.

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Step 3.4.2.1.5.1

Multiply $- 8$ by $3$ .

$0 = \frac{- 24 + 16}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.5.2

Add $- 24$ and $16$ .

$0 = \frac{- 8}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = \frac{- 8}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.6

Move the negative in front of the fraction.

$0 = - \frac{8}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.7

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

$0 = - \frac{8}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.8

Combine $- 3 c$ and $\frac{3}{3}$ .

$0 = - \frac{8}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.9

Combine the numerators over the common denominator.

$0 = - \frac{8}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.10

Combine the numerators over the common denominator.

$0 = \frac{- 8 + 8 c - 3 c \cdot 3}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.11

Multiply $3$ by $- 3$ .

$0 = \frac{- 8 + 8 c - 9 c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.12

Subtract $9 c$ from $8 c$ .

$0 = \frac{- 8 - c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.13

Rewrite $- 8$ as $- 1 (8)$ .

$0 = \frac{- 1 \cdot 8 - c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.14

Factor $- 1$ out of $- c$ .

$0 = \frac{- 1 \cdot 8 - (c)}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.15

Factor $- 1$ out of $- 1 (8) - (c)$ .

$0 = \frac{- 1 (8 + c)}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.2.1.16

Move the negative in front of the fraction.

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Step 3.4.3

Replace all occurrences of $b$ in $a = - 2 - b - c$ with $- \frac{8}{3} - \frac{4 c}{3}$ .

$a = - 2 - (- \frac{8}{3} - \frac{4 c}{3}) - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $- 2 - (- \frac{8}{3} - \frac{4 c}{3}) - c$ .

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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.

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.2

Multiply $- - \frac{8}{3}$ .

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Step 3.4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$a = - 2 + 1 (\frac{8}{3}) + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.2.2

Multiply $\frac{8}{3}$ by $1$ .

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.3

Multiply $- - \frac{4 c}{3}$ .

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Step 3.4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$a = - 2 + \frac{8}{3} + 1 (\frac{4 c}{3}) - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.3.2

Multiply $\frac{4 c}{3}$ by $1$ .

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - 2 \cdot \frac{3}{3} + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.3

Combine $- 2$ and $\frac{3}{3}$ .

$a = \frac{- 2 \cdot 3}{3} + \frac{8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$a = \frac{- 2 \cdot 3 + 8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $- 2$ by $3$ .

$a = \frac{- 6 + 8}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.5.2

Add $- 6$ and $8$ .

$a = \frac{2}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = \frac{2}{3} + \frac{4 c}{3} - c$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.6

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

$a = \frac{2}{3} + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.7

Combine $- c$ and $\frac{3}{3}$ .

$a = \frac{2}{3} + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.8

Combine the numerators over the common denominator.

$a = \frac{2}{3} + \frac{4 c - c \cdot 3}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.9

Combine the numerators over the common denominator.

$a = \frac{2 + 4 c - c \cdot 3}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.10

Multiply $3$ by $- 1$ .

$a = \frac{2 + 4 c - 3 c}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.4.4.1.11

Subtract $3 c$ from $4 c$ .

$a = \frac{2 + c}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = \frac{2 + c}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = \frac{2 + c}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = \frac{2 + c}{3}$

$0 = - \frac{8 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.5

Solve for $c$ in $0 = - \frac{8 + c}{3}$ .

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Step 3.5.1

Set the numerator equal to zero.

$8 + c = 0$

$a = \frac{2 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.5.2

Subtract $8$ from both sides of the equation.

$c = - 8$

$a = \frac{2 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$c = - 8$

$a = \frac{2 + c}{3}$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.6

Replace all occurrences of $c$ with $- 8$ in each equation.

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Step 3.6.1

Replace all occurrences of $c$ in $a = \frac{2 + c}{3}$ with $- 8$ .

$a = \frac{2 - 8}{3}$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.6.2

Simplify $a = \frac{2 - 8}{3}$ .

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Step 3.6.2.1

Simplify the left side.

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Step 3.6.2.1.1

Remove parentheses.

$a = \frac{2 - 8}{3}$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = \frac{2 - 8}{3}$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.6.2.2

Simplify the right side.

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Step 3.6.2.2.1

Simplify $\frac{2 - 8}{3}$ .

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Step 3.6.2.2.1.1

Subtract $8$ from $2$ .

$a = \frac{- 6}{3}$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.6.2.2.1.2

Divide $- 6$ by $3$ .

$a = - 2$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

$a = - 2$

$c = - 8$

$b = - \frac{8}{3} - \frac{4 c}{3}$

Step 3.6.3

Replace all occurrences of $c$ in $b = - \frac{8}{3} - \frac{4 c}{3}$ with $- 8$ .

$b = - \frac{8}{3} - \frac{4 (- 8)}{3}$

$a = - 2$

$c = - 8$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- \frac{8}{3} - \frac{4 (- 8)}{3}$ .

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Step 3.6.4.1.1

Combine the numerators over the common denominator.

$b = \frac{- 8 - 4 \cdot - 8}{3}$

$a = - 2$

$c = - 8$

Step 3.6.4.1.2

Simplify the expression.

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Step 3.6.4.1.2.1

Multiply $- 4$ by $- 8$ .

$b = \frac{- 8 + 32}{3}$

$a = - 2$

$c = - 8$

Step 3.6.4.1.2.2

Add $- 8$ and $32$ .

$b = \frac{24}{3}$

$a = - 2$

$c = - 8$

Step 3.6.4.1.2.3

Divide $24$ by $3$ .

$b = 8$

$a = - 2$

$c = - 8$

$b = 8$

$a = - 2$

$c = - 8$

$b = 8$

$a = - 2$

$c = - 8$

$b = 8$

$a = - 2$

$c = - 8$

$b = 8$

$a = - 2$

$c = - 8$

Step 3.7

List all of the solutions.

$b = 8, a = - 2, c = - 8$

Step 4

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

$y = - 2 x^{2} + 8 x - 8$

Step 5