Algebra Examples

$(- 6, 0)$ , $(- 4, 0)$ , $(- 3, - 18)$

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.

$0 = a {(- 6)}^{2} + b (- 6) + c, 0 = a {(- 4)}^{2} + b (- 4) + c, - 18 = a {(- 3)}^{2} + b (- 3) + c$

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 $0 = a {(- 6)}^{2} + b (- 6) + c$ .

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Step 3.1.1

Rewrite the equation as $a {(- 6)}^{2} + b (- 6) + c = 0$ .

$a {(- 6)}^{2} + b (- 6) + c = 0$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Raise $- 6$ to the power of $2$ .

$a \cdot 36 + b (- 6) + c = 0$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.1.2.2

Move $36$ to the left of $a$ .

$36 \cdot a + b (- 6) + c = 0$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.1.2.3

Move $- 6$ to the left of $b$ .

$36 a - 6 b + c = 0$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$36 a - 6 b + c = 0$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

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 $36 a$ from both sides of the equation.

$- 6 b + c = - 36 a$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.1.3.2

Add $6 b$ to both sides of the equation.

$c = - 36 a + 6 b$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$c = - 36 a + 6 b$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$c = - 36 a + 6 b$

$0 = a {(- 4)}^{2} + b (- 4) + c$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2

Replace all occurrences of $c$ with $- 36 a + 6 b$ in each equation.

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Step 3.2.1

Replace all occurrences of $c$ in $0 = a {(- 4)}^{2} + b (- 4) + c$ with $- 36 a + 6 b$ .

$0 = a {(- 4)}^{2} + b (- 4) - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2

Simplify $0 = a {(- 4)}^{2} + b (- 4) - 36 a + 6 b$ .

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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.

$0 = a {(- 4)}^{2} + b (- 4) - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = a {(- 4)}^{2} + b (- 4) - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify $a {(- 4)}^{2} + b (- 4) - 36 a + 6 b$ .

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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 $- 4$ to the power of $2$ .

$0 = a \cdot 16 + b (- 4) - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2.2.1.1.2

Move $16$ to the left of $a$ .

$0 = 16 \cdot a + b (- 4) - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2.2.1.1.3

Move $- 4$ to the left of $b$ .

$0 = 16 a - 4 b - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = 16 a - 4 b - 36 a + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.2.1.2.1

Subtract $36 a$ from $16 a$ .

$0 = - 20 a - 4 b + 6 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.2.2.1.2.2

Add $- 4 b$ and $6 b$ .

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) + c$

Step 3.2.3

Replace all occurrences of $c$ in $- 18 = a {(- 3)}^{2} + b (- 3) + c$ with $- 36 a + 6 b$ .

$- 18 = a {(- 3)}^{2} + b (- 3) - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4

Simplify $- 18 = a {(- 3)}^{2} + b (- 3) - 36 a + 6 b$ .

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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.

$- 18 = a {(- 3)}^{2} + b (- 3) - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = a {(- 3)}^{2} + b (- 3) - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify $a {(- 3)}^{2} + b (- 3) - 36 a + 6 b$ .

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

$- 18 = a \cdot 9 + b (- 3) - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4.2.1.1.2

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

$- 18 = 9 \cdot a + b (- 3) - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4.2.1.1.3

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

$- 18 = 9 a - 3 b - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = 9 a - 3 b - 36 a + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4.2.1.2

Simplify by adding terms.

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Step 3.2.4.2.1.2.1

Subtract $36 a$ from $9 a$ .

$- 18 = - 27 a - 3 b + 6 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.2.4.2.1.2.2

Add $- 3 b$ and $6 b$ .

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$- 18 = - 27 a + 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3

Solve for $a$ in $- 18 = - 27 a + 3 b$ .

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Step 3.3.1

Rewrite the equation as $- 27 a + 3 b = - 18$ .

$- 27 a + 3 b = - 18$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.2

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

$- 27 a = - 18 - 3 b$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3

Divide each term in $- 27 a = - 18 - 3 b$ by $- 27$ and simplify.

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Step 3.3.3.1

Divide each term in $- 27 a = - 18 - 3 b$ by $- 27$ .

$\frac{- 27 a}{- 27} = \frac{- 18}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 27$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 27 a}{- 27} = \frac{- 18}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 18}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{- 18}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{- 18}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

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

Cancel the common factor of $- 18$ and $- 27$ .

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Step 3.3.3.3.1.1.1

Factor $- 9$ out of $- 18$ .

$a = \frac{- 9 \cdot 2}{- 27} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

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 $- 9$ out of $- 27$ .

$a = \frac{- 9 \cdot 2}{- 9 \cdot 3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{- 9 \cdot 2}{- 9 \cdot 3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{2}{3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.3.1.2

Cancel the common factor of $- 3$ and $- 27$ .

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Step 3.3.3.3.1.2.1

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

$a = \frac{2}{3} + \frac{- 3 b}{- 27}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

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 $- 3$ out of $- 27$ .

$a = \frac{2}{3} + \frac{- 3 b}{- 3 \cdot 9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.3.1.2.2.2

Cancel the common factor.

$a = \frac{2}{3} + \frac{- 3 b}{- 3 \cdot 9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.3.3.3.1.2.2.3

Rewrite the expression.

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

$a = \frac{2}{3} + \frac{b}{9}$

$0 = - 20 a + 2 b$

$c = - 36 a + 6 b$

Step 3.4

Replace all occurrences of $a$ with $\frac{2}{3} + \frac{b}{9}$ in each equation.

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Step 3.4.1

Replace all occurrences of $a$ in $0 = - 20 a + 2 b$ with $\frac{2}{3} + \frac{b}{9}$ .

$0 = - 20 (\frac{2}{3} + \frac{b}{9}) + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 20 (\frac{2}{3} + \frac{b}{9}) + 2 b$ .

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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.

$0 = - 20 (\frac{2}{3}) - 20 \frac{b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.1.2

Multiply $- 20 (\frac{2}{3})$ .

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Step 3.4.2.1.1.2.1

Combine $- 20$ and $\frac{2}{3}$ .

$0 = \frac{- 20 \cdot 2}{3} - 20 \frac{b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.1.2.2

Multiply $- 20$ by $2$ .

$0 = \frac{- 40}{3} - 20 \frac{b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = \frac{- 40}{3} - 20 \frac{b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.1.3

Combine $- 20$ and $\frac{b}{9}$ .

$0 = \frac{- 40}{3} + \frac{- 20 b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.1.4

Simplify each term.

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Step 3.4.2.1.1.4.1

Move the negative in front of the fraction.

$0 = - \frac{40}{3} + \frac{- 20 b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.1.4.2

Move the negative in front of the fraction.

$0 = - \frac{40}{3} - \frac{20 b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} - \frac{20 b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} - \frac{20 b}{9} + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.2

To write $2 b$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$0 = - \frac{40}{3} - \frac{20 b}{9} + 2 b \cdot \frac{9}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.3

Simplify terms.

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Step 3.4.2.1.3.1

Combine $2 b$ and $\frac{9}{9}$ .

$0 = - \frac{40}{3} - \frac{20 b}{9} + \frac{2 b \cdot 9}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.3.2

Combine the numerators over the common denominator.

$0 = - \frac{40}{3} + \frac{- 20 b + 2 b \cdot 9}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} + \frac{- 20 b + 2 b \cdot 9}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4

Simplify each term.

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Step 3.4.2.1.4.1

Simplify the numerator.

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Step 3.4.2.1.4.1.1

Factor $2 b$ out of $- 20 b + 2 b \cdot 9$ .

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Step 3.4.2.1.4.1.1.1

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

$0 = - \frac{40}{3} + \frac{2 b (- 10) + 2 b \cdot 9}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4.1.1.2

Factor $2 b$ out of $2 b \cdot 9$ .

$0 = - \frac{40}{3} + \frac{2 b (- 10) + 2 b (9)}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4.1.1.3

Factor $2 b$ out of $2 b (- 10) + 2 b (9)$ .

$0 = - \frac{40}{3} + \frac{2 b (- 10 + 9)}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} + \frac{2 b (- 10 + 9)}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4.1.2

Add $- 10$ and $9$ .

$0 = - \frac{40}{3} + \frac{2 b \cdot - 1}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4.1.3

Multiply $- 1$ by $2$ .

$0 = - \frac{40}{3} + \frac{- 2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} + \frac{- 2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.2.1.4.2

Move the negative in front of the fraction.

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 36 a + 6 b$

Step 3.4.3

Replace all occurrences of $a$ in $c = - 36 a + 6 b$ with $\frac{2}{3} + \frac{b}{9}$ .

$c = - 36 (\frac{2}{3} + \frac{b}{9}) + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $- 36 (\frac{2}{3} + \frac{b}{9}) + 6 b$ .

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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.

$c = - 36 (\frac{2}{3}) - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.4.1.1.2.1

Factor $3$ out of $- 36$ .

$c = 3 (- 12) (\frac{2}{3}) - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.2.2

Cancel the common factor.

$c = 3 \cdot (- 12 (\frac{2}{3})) - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.2.3

Rewrite the expression.

$c = - 12 \cdot 2 - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 12 \cdot 2 - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.3

Multiply $- 12$ by $2$ .

$c = - 24 - 36 \frac{b}{9} + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.4

Cancel the common factor of $9$ .

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Step 3.4.4.1.1.4.1

Factor $9$ out of $- 36$ .

$c = - 24 + 9 (- 4) (\frac{b}{9}) + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.4.2

Cancel the common factor.

$c = - 24 + 9 \cdot (- 4 \frac{b}{9}) + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.1.4.3

Rewrite the expression.

$c = - 24 - 4 b + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 24 - 4 b + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 24 - 4 b + 6 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.4.4.1.2

Add $- 4 b$ and $6 b$ .

$c = - 24 + 2 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 24 + 2 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 24 + 2 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 24 + 2 b$

$0 = - \frac{40}{3} - \frac{2 b}{9}$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5

Solve for $b$ in $0 = - \frac{40}{3} - \frac{2 b}{9}$ .

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Step 3.5.1

Rewrite the equation as $- \frac{40}{3} - \frac{2 b}{9} = 0$ .

$- \frac{40}{3} - \frac{2 b}{9} = 0$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.2

Add $\frac{40}{3}$ to both sides of the equation.

$- \frac{2 b}{9} = \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.3

Multiply both sides of the equation by $- \frac{9}{2}$ .

$- \frac{9}{2} \cdot (- \frac{2 b}{9}) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4

Simplify both sides of the equation.

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Step 3.5.4.1

Simplify the left side.

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Step 3.5.4.1.1

Simplify $- \frac{9}{2} (- \frac{2 b}{9})$ .

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Step 3.5.4.1.1.1

Cancel the common factor of $9$ .

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Step 3.5.4.1.1.1.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$\frac{- 9}{2} \cdot (- \frac{2 b}{9}) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.1.2

Move the leading negative in $- \frac{2 b}{9}$ into the numerator.

$\frac{- 9}{2} \cdot \frac{- 2 b}{9} = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.1.3

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{2} \cdot \frac{- 2 b}{9} = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.1.4

Cancel the common factor.

$\frac{9 \cdot - 1}{2} \cdot \frac{- 2 b}{9} = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} \cdot (- 2 b) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$\frac{- 1}{2} \cdot (- 2 b) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.4.1.1.2.1

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

$\frac{- 1}{2} \cdot (2 (- b)) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (- b)) = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.2.3

Rewrite the expression.

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.3

Multiply.

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Step 3.5.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.1.1.3.2

Multiply $b$ by $1$ .

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - \frac{9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2

Simplify the right side.

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Step 3.5.4.2.1

Simplify $- \frac{9}{2} \cdot \frac{40}{3}$ .

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Step 3.5.4.2.1.1

Cancel the common factor of $3$ .

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Step 3.5.4.2.1.1.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$b = \frac{- 9}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.1.2

Factor $3$ out of $- 9$ .

$b = \frac{3 (- 3)}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.1.3

Cancel the common factor.

$b = \frac{3 \cdot - 3}{2} \cdot \frac{40}{3}$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.1.4

Rewrite the expression.

$b = \frac{- 3}{2} \cdot 40$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = \frac{- 3}{2} \cdot 40$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.2

Cancel the common factor of $2$ .

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Step 3.5.4.2.1.2.1

Factor $2$ out of $40$ .

$b = \frac{- 3}{2} \cdot (2 (20))$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.2.2

Cancel the common factor.

$b = \frac{- 3}{2} \cdot (2 \cdot 20)$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.2.3

Rewrite the expression.

$b = - 3 \cdot 20$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - 3 \cdot 20$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.5.4.2.1.3

Multiply $- 3$ by $20$ .

$b = - 60$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - 60$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - 60$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - 60$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

$b = - 60$

$c = - 24 + 2 b$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.6

Replace all occurrences of $b$ with $- 60$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $c = - 24 + 2 b$ with $- 60$ .

$c = - 24 + 2 (- 60)$

$b = - 60$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- 24 + 2 (- 60)$ .

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Step 3.6.2.1.1

Multiply $2$ by $- 60$ .

$c = - 24 - 120$

$b = - 60$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.6.2.1.2

Subtract $120$ from $- 24$ .

$c = - 144$

$b = - 60$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 144$

$b = - 60$

$a = \frac{2}{3} + \frac{b}{9}$

$c = - 144$

$b = - 60$

$a = \frac{2}{3} + \frac{b}{9}$

Step 3.6.3

Replace all occurrences of $b$ in $a = \frac{2}{3} + \frac{b}{9}$ with $- 60$ .

$a = \frac{2}{3} + \frac{- 60}{9}$

$c = - 144$

$b = - 60$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $\frac{2}{3} + \frac{- 60}{9}$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Cancel the common factor of $- 60$ and $9$ .

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Step 3.6.4.1.1.1.1

Factor $3$ out of $- 60$ .

$a = \frac{2}{3} + \frac{3 (- 20)}{9}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.1.1.2

Cancel the common factors.

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Step 3.6.4.1.1.1.2.1

Factor $3$ out of $9$ .

$a = \frac{2}{3} + \frac{3 \cdot - 20}{3 \cdot 3}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.1.1.2.2

Cancel the common factor.

$a = \frac{2}{3} + \frac{3 \cdot - 20}{3 \cdot 3}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.1.1.2.3

Rewrite the expression.

$a = \frac{2}{3} + \frac{- 20}{3}$

$c = - 144$

$b = - 60$

$a = \frac{2}{3} + \frac{- 20}{3}$

$c = - 144$

$b = - 60$

$a = \frac{2}{3} + \frac{- 20}{3}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.1.2

Move the negative in front of the fraction.

$a = \frac{2}{3} - \frac{20}{3}$

$c = - 144$

$b = - 60$

$a = \frac{2}{3} - \frac{20}{3}$

$c = - 144$

$b = - 60$

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.

$a = \frac{2 - 20}{3}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.2.2

Simplify the expression.

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Step 3.6.4.1.2.2.1

Subtract $20$ from $2$ .

$a = \frac{- 18}{3}$

$c = - 144$

$b = - 60$

Step 3.6.4.1.2.2.2

Divide $- 18$ by $3$ .

$a = - 6$

$c = - 144$

$b = - 60$

$a = - 6$

$c = - 144$

$b = - 60$

$a = - 6$

$c = - 144$

$b = - 60$

$a = - 6$

$c = - 144$

$b = - 60$

$a = - 6$

$c = - 144$

$b = - 60$

$a = - 6$

$c = - 144$

$b = - 60$

Step 3.7

List all of the solutions.

$a = - 6, c = - 144, b = - 60$

Step 4

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

$y = - 6 x^{2} - 60 x - 144$

Step 5