Finite Math Examples

(2, 0)

$(2, 0)$ ,

(3, - 2)

$(3, - 2)$ ,

(1, - 2)

$(1, - 2)$

Step 1

Use the standard form of a quadratic equation

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

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

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

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

Step 2

Create a system of equations by substituting the

x

$x$ and

y

$y$ 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

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

Step 3

Solve the system of equations to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

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Step 3.1

Solve for

a

$a$ in

- 2 = a + b + c

$- 2 = a + b + c$ .

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Step 3.1.1

Rewrite the equation as

a + b + c = - 2

$a + b + c = - 2$ .

a + b + c = - 2

$a + b + c = - 2$

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

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

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

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

Step 3.1.2

Move all terms not containing

a

$a$ to the right side of the equation.

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Step 3.1.2.1

Subtract

b

$b$ from both sides of the equation.

a + c = - 2 - b

$a + c = - 2 - b$

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

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

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

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

Step 3.1.2.2

Subtract

c

$c$ from both sides of the equation.

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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

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

Step 3.2

Replace all occurrences of

a

$a$ with

- 2 - b - c

$- 2 - b - c$ in each equation.

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Step 3.2.1

Replace all occurrences of

a

$a$ in

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

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

- 2 - b - c

$- 2 - b - c$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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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

$2$ to the power of

2

$2$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

$- 2$ by

4

$4$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

Step 3.2.2.1.1.3.2

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

Step 3.2.2.1.1.3.3

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

Step 3.2.2.1.1.4

Move

2

$2$ to the left of

b

$b$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

$- 4 b$ and

2 b

$2 b$ .

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

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

Step 3.2.2.1.2.2

Add

- 4 c

$- 4 c$ and

c

$c$ .

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

Step 3.2.3

Replace all occurrences of

a

$a$ in

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

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

- 2 - b - c

$- 2 - b - c$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - 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

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

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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

$3$ to the power of

2

$2$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - 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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - 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

$- 2$ by

9

$9$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.2.4.1.1.3.2

Multiply

9

$9$ by

- 1

$- 1$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.2.4.1.1.3.3

Multiply

9

$9$ by

- 1

$- 1$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.2.4.1.1.4

Move

3

$3$ to the left of

b

$b$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - 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

$- 9 b$ and

3 b

$3 b$ .

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

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.2.4.1.2.2

Add

- 9 c

$- 9 c$ and

c

$c$ .

- 2 = - 18 - 6 b - 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

- 2 = - 18 - 6 b - 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

- 2 = - 18 - 6 b - 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

- 2 = - 18 - 6 b - 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

- 2 = - 18 - 6 b - 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.3

Solve for

b

$b$ in

- 2 = - 18 - 6 b - 8 c

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

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Step 3.3.1

Rewrite the equation as

- 18 - 6 b - 8 c = - 2

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

- 18 - 6 b - 8 c = - 2

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.3.2

Move all terms not containing

b

$b$ to the right side of the equation.

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Step 3.3.2.1

Add

18

$18$ to both sides of the equation.

- 6 b - 8 c = - 2 + 18

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.3.2.2

Add

8 c

$8 c$ to both sides of the equation.

- 6 b = - 2 + 18 + 8 c

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

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.3.2.3

Add

- 2

$- 2$ and

18

$18$ .

- 6 b = 16 + 8 c

$- 6 b = 16 + 8 c$

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

- 6 b = 16 + 8 c

$- 6 b = 16 + 8 c$

0 = - 8 - 2 b - 3 c

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

a = - 2 - b - c

$a = - 2 - b - c$

Step 3.3.3

Divide each term in

- 6 b = 16 + 8 c

$- 6 b = 16 + 8 c$ by

- 6

$- 6$ and simplify.

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Step 3.3.3.1

Divide each term in

- 6 b = 16 + 8 c

$- 6 b = 16 + 8 c$ by

- 6

$- 6$ .

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

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

0 = - 8 - 2 b - 3 c

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

$a = - 2 - b - c$

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