Algebra Examples

$(1, - 1)$ , $(2, - 5)$ , $(3, - 7)$

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.

$- 1 = a {(1)}^{2} + b (1) + c, - 5 = a {(2)}^{2} + b (2) + c, - 7 = a {(3)}^{2} + b (3) + c$

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 $a$ in $- 1 = a + b + c$ .

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Step 3.1.1

Rewrite the equation as $a + b + c = - 1$ .

$a + b + c = - 1$

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

$- 7 = a \cdot 3^{2} + b (3) + c$

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.

$a + c = - 1 - b$

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

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.1.2.2

Subtract $c$ from both sides of the equation.

$a = - 1 - b - c$

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

$- 7 = a \cdot 3^{2} + b (3) + c$

$a = - 1 - b - c$

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

$- 7 = a \cdot 3^{2} + b (3) + c$

$a = - 1 - b - c$

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

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2

Replace all occurrences of $a$ with $- 1 - b - c$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $- 5 = a \cdot 2^{2} + b (2) + c$ with $- 1 - b - c$ .

$- 5 = (- 1 - b - c) \cdot 2^{2} + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(- 1 - b - c) \cdot 2^{2} + b (2) + c$ .

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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 $2$ .

$- 5 = (- 1 - b - c) \cdot 4 + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.1.2

Apply the distributive property.

$- 5 = - 1 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.1.3

Simplify.

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Step 3.2.2.1.1.3.1

Multiply $- 1$ by $4$ .

$- 5 = - 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.1.3.2

Multiply $4$ by $- 1$ .

$- 5 = - 4 - 4 b - c \cdot 4 + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.1.3.3

Multiply $4$ by $- 1$ .

$- 5 = - 4 - 4 b - 4 c + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 4 b - 4 c + b (2) + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.1.4

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

$- 5 = - 4 - 4 b - 4 c + 2 b + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 4 b - 4 c + 2 b + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

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 $2 b$ .

$- 5 = - 4 - 2 b - 4 c + c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.1.2.2

Add $- 4 c$ and $c$ .

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.3

Replace all occurrences of $a$ in $- 7 = a \cdot 3^{2} + b (3) + c$ with $- 1 - b - c$ .

$- 7 = (- 1 - b - c) \cdot 3^{2} + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $(- 1 - b - c) \cdot 3^{2} + b (3) + c$ .

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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 $2$ .

$- 7 = (- 1 - b - c) \cdot 9 + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.1.2

Apply the distributive property.

$- 7 = - 1 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.1.3

Simplify.

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Step 3.2.4.1.1.3.1

Multiply $- 1$ by $9$ .

$- 7 = - 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.1.3.2

Multiply $9$ by $- 1$ .

$- 7 = - 9 - 9 b - c \cdot 9 + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.1.3.3

Multiply $9$ by $- 1$ .

$- 7 = - 9 - 9 b - 9 c + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 9 b - 9 c + b (3) + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.1.4

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

$- 7 = - 9 - 9 b - 9 c + 3 b + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 9 b - 9 c + 3 b + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $3 b$ .

$- 7 = - 9 - 6 b - 9 c + c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.2.4.1.2.2

Add $- 9 c$ and $c$ .

$- 7 = - 9 - 6 b - 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 6 b - 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 6 b - 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 6 b - 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 7 = - 9 - 6 b - 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3

Solve for $b$ in $- 7 = - 9 - 6 b - 8 c$ .

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Step 3.3.1

Rewrite the equation as $- 9 - 6 b - 8 c = - 7$ .

$- 9 - 6 b - 8 c = - 7$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $9$ to both sides of the equation.

$- 6 b - 8 c = - 7 + 9$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.2.2

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

$- 6 b = - 7 + 9 + 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.2.3

Add $- 7$ and $9$ .

$- 6 b = 2 + 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$- 6 b = 2 + 8 c$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3

Divide each term in $- 6 b = 2 + 8 c$ by $- 6$ and simplify.

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Step 3.3.3.1

Divide each term in $- 6 b = 2 + 8 c$ by $- 6$ .

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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.

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.2.1.2

Divide $b$ by $1$ .

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $2$ and $- 6$ .

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Step 3.3.3.3.1.1.1

Factor $2$ out of $2$ .

$b = \frac{2 (1)}{- 6} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $- 6$ .

$b = \frac{2 \cdot 1}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$b = \frac{2 \cdot 1}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$b = \frac{1}{- 3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = \frac{1}{- 3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = \frac{1}{- 3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$b = - \frac{1}{3} + \frac{8 c}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $8 c$ .

$b = - \frac{1}{3} + \frac{2 (4 c)}{- 6}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

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 $- 6$ .

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

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$b = - \frac{1}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$b = - \frac{1}{3} + \frac{4 c}{- 3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} + \frac{4 c}{- 3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} + \frac{4 c}{- 3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.3.3.3.1.4

Move the negative in front of the fraction.

$b = - \frac{1}{3} - \frac{4 c}{3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$- 5 = - 4 - 2 b - 3 c$

$a = - 1 - b - c$

Step 3.4

Replace all occurrences of $b$ with $- \frac{1}{3} - \frac{4 c}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $- 5 = - 4 - 2 b - 3 c$ with $- \frac{1}{3} - \frac{4 c}{3}$ .

$- 5 = - 4 - 2 (- \frac{1}{3} - \frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 4 - 2 (- \frac{1}{3} - \frac{4 c}{3}) - 3 c$ .

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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.

$- 5 = - 4 - 2 (- \frac{1}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.1.2

Multiply $- 2 (- \frac{1}{3})$ .

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Step 3.4.2.1.1.2.1

Multiply $- 1$ by $- 2$ .

$- 5 = - 4 + 2 (\frac{1}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.1.2.2

Combine $2$ and $\frac{1}{3}$ .

$- 5 = - 4 + \frac{2}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = - 4 + \frac{2}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

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 $- 2$ .

$- 5 = - 4 + \frac{2}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.1.3.2

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

$- 5 = - 4 + \frac{2}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.1.3.3

Multiply $4$ by $2$ .

$- 5 = - 4 + \frac{2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = - 4 + \frac{2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = - 4 + \frac{2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 5 = - 4 \cdot \frac{3}{3} + \frac{2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.3

Combine $- 4$ and $\frac{3}{3}$ .

$- 5 = \frac{- 4 \cdot 3}{3} + \frac{2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$- 5 = \frac{- 4 \cdot 3 + 2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.5

Simplify the numerator.

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Step 3.4.2.1.5.1

Multiply $- 4$ by $3$ .

$- 5 = \frac{- 12 + 2}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.5.2

Add $- 12$ and $2$ .

$- 5 = \frac{- 10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = \frac{- 10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.6

Move the negative in front of the fraction.

$- 5 = - \frac{10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.7

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

$- 5 = - \frac{10}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.8

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

$- 5 = - \frac{10}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.9

Combine the numerators over the common denominator.

$- 5 = - \frac{10}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.10

Combine the numerators over the common denominator.

$- 5 = \frac{- 10 + 8 c - 3 c \cdot 3}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.11

Multiply $3$ by $- 3$ .

$- 5 = \frac{- 10 + 8 c - 9 c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.12

Subtract $9 c$ from $8 c$ .

$- 5 = \frac{- 10 - c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.13

Rewrite $- 10$ as $- 1 (10)$ .

$- 5 = \frac{- 1 \cdot 10 - c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.14

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

$- 5 = \frac{- 1 \cdot 10 - (c)}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.15

Factor $- 1$ out of $- 1 (10) - (c)$ .

$- 5 = \frac{- 1 (10 + c)}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.2.1.16

Move the negative in front of the fraction.

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 3.4.3

Replace all occurrences of $b$ in $a = - 1 - b - c$ with $- \frac{1}{3} - \frac{4 c}{3}$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $- 1 - (- \frac{1}{3} - \frac{4 c}{3}) - c$ .

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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.

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.2

Multiply $- - \frac{1}{3}$ .

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Step 3.4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.2.2

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

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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 $- 1$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.1.3.2

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

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = - 1 + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - 1 \cdot \frac{3}{3} + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.3

Combine $- 1$ and $\frac{3}{3}$ .

$a = \frac{- 1 \cdot 3}{3} + \frac{1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$a = \frac{- 1 \cdot 3 + 1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $- 1$ by $3$ .

$a = \frac{- 3 + 1}{3} + \frac{4 c}{3} - c$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.5.2

Add $- 3$ and $1$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.6

Move the negative in front of the fraction.

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.7

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

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.8

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

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.9

Combine the numerators over the common denominator.

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.10

Combine the numerators over the common denominator.

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.11

Multiply $3$ by $- 1$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.12

Subtract $3 c$ from $4 c$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.13

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

$a = \frac{- 1 \cdot 2 + c}{3}$

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.14

Factor $- 1$ out of $c$ .

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.15

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

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.4.4.1.16

Move the negative in front of the fraction.

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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

$- 5 = - \frac{10 + c}{3}$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5

Solve for $c$ in $- 5 = - \frac{10 + c}{3}$ .

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Step 3.5.1

Rewrite the equation as $- \frac{10 + c}{3} = - 5$ .

$- \frac{10 + c}{3} = - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.2

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{10 + c}{3}) = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3

Simplify both sides of the equation.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Simplify $- 3 (- \frac{10 + c}{3})$ .

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Step 3.5.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.3.1.1.1.1

Move the leading negative in $- \frac{10 + c}{3}$ into the numerator.

$- 3 \frac{- (10 + c)}{3} = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (10 + c)}{3}) = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (10 + c)}{3}) = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.1.1.1.4

Rewrite the expression.

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.1.1.2

Multiply.

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Step 3.5.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (10 + c) = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.1.1.2.2

Multiply $10 + c$ by $1$ .

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Multiply $- 3$ by $- 5$ .

$10 + c = 15$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = 15$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$10 + c = 15$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.4

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

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Step 3.5.4.1

Subtract $10$ from both sides of the equation.

$c = 15 - 10$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.5.4.2

Subtract $10$ from $15$ .

$c = 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$c = 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

$c = 5$

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

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6

Replace all occurrences of $c$ with $5$ in each equation.

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Step 3.6.1

Replace all occurrences of $c$ in $a = - \frac{2 - c}{3}$ with $5$ .

$a = - \frac{2 - (5)}{3}$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

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

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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 $- 1$ by $5$ .

$a = - \frac{2 - 5}{3}$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6.2.1.1.2

Subtract $5$ from $2$ .

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

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

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

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6.2.1.2

Simplify the expression.

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Step 3.6.2.1.2.1

Divide $- 3$ by $3$ .

$a = 1$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6.2.1.2.2

Multiply $- 1$ by $- 1$ .

$a = 1$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = 1$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = 1$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

$a = 1$

$c = 5$

$b = - \frac{1}{3} - \frac{4 c}{3}$

Step 3.6.3

Replace all occurrences of $c$ in $b = - \frac{1}{3} - \frac{4 c}{3}$ with $5$ .

$b = - \frac{1}{3} - \frac{4 (5)}{3}$

$a = 1$

$c = 5$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- \frac{1}{3} - \frac{4 (5)}{3}$ .

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Step 3.6.4.1.1

Combine the numerators over the common denominator.

$b = \frac{- 1 - 4 \cdot 5}{3}$

$a = 1$

$c = 5$

Step 3.6.4.1.2

Simplify the expression.

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Step 3.6.4.1.2.1

Multiply $- 4$ by $5$ .

$b = \frac{- 1 - 20}{3}$

$a = 1$

$c = 5$

Step 3.6.4.1.2.2

Subtract $20$ from $- 1$ .

$b = \frac{- 21}{3}$

$a = 1$

$c = 5$

Step 3.6.4.1.2.3

Divide $- 21$ by $3$ .

$b = - 7$

$a = 1$

$c = 5$

$b = - 7$

$a = 1$

$c = 5$

$b = - 7$

$a = 1$

$c = 5$

$b = - 7$

$a = 1$

$c = 5$

$b = - 7$

$a = 1$

$c = 5$

Step 3.7

List all of the solutions.

$b = - 7, a = 1, c = 5$

Step 4

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

$y = x^{2} - 7 x + 5$

Step 5