Algebra Examples

$(1, 0)$ , $(4, 2)$ , $(3, - 7)$

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.

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

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.

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

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

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

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

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Simplify each equation.

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Remove parentheses.

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

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

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

Simplify each term.

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One to any power is one.

$0 = a \cdot 1 + b (1) + c$

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

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

Multiply $a$ by $1$ .

$0 = a + b (1) + c$

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

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

Multiply $b$ by $1$ .

$0 = a + b + c$

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

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

$0 = a + b + c$

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

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

Remove parentheses.

$0 = a + b + c$

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

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

Simplify each term.

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

$0 = a + b + c$

$2 = a \cdot 16 + b (4) + c$

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

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

$0 = a + b + c$

$2 = 16 \cdot a + b (4) + c$

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

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

Remove parentheses.

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

Simplify each term.

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

$0 = a + b + c$

$2 = 16 a + 4 b + c$

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

Solve for $a$ in the first equation.

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Rewrite the equation as $a + b + c = 0$ .

$a + b + c = 0$

$2 = 16 a + 4 b + c$

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

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

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Subtract $b$ from both sides of the equation.

$a + c = - b$

$2 = 16 a + 4 b + c$

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

Subtract $c$ from both sides of the equation.

$a = - b - c$

$2 = 16 a + 4 b + c$

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

$a = - b - c$

$2 = 16 a + 4 b + c$

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

$a = - b - c$

$2 = 16 a + 4 b + c$

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

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

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Replace all occurrences of $a$ in $2 = 16 a + 4 b + c$ with $- b - c$ .

$a = - b - c$

$2 = 16 (- b - c) + 4 b + c$

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

Replace all occurrences of $a$ in $- 7 = 9 a + 3 b + c$ with $- b - c$ .

$a = - b - c$

$2 = 16 (- b - c) + 4 b + c$

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

$a = - b - c$

$2 = 16 (- b - c) + 4 b + c$

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

Simplify each equation.

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Simplify $16 (- b - c) + 4 b + c$ .

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Simplify each term.

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Apply the distributive property.

$a = - b - c$

$2 = 16 (- b) + 16 (- c) + 4 b + c$

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

Multiply $- 1$ by $16$ .

$a = - b - c$

$2 = - 16 b + 16 (- c) + 4 b + c$

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

Multiply $- 1$ by $16$ .

$a = - b - c$

$2 = - 16 b - 16 c + 4 b + c$

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

$a = - b - c$

$2 = - 16 b - 16 c + 4 b + c$

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

Simplify by adding terms.

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Add $- 16 b$ and $4 b$ .

$a = - b - c$

$2 = - 12 b - 16 c + c$

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

Add $- 16 c$ and $c$ .

$a = - b - c$

$2 = - 12 b - 15 c$

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

$a = - b - c$

$2 = - 12 b - 15 c$

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

$a = - b - c$

$2 = - 12 b - 15 c$

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

Simplify $9 (- b - c) + 3 b + c$ .

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Simplify each term.

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Apply the distributive property.

$a = - b - c$

$2 = - 12 b - 15 c$

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

Multiply $- 1$ by $9$ .

$a = - b - c$

$2 = - 12 b - 15 c$

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

Multiply $- 1$ by $9$ .

$a = - b - c$

$2 = - 12 b - 15 c$

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

$a = - b - c$

$2 = - 12 b - 15 c$

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

Simplify by adding terms.

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

$a = - b - c$

$2 = - 12 b - 15 c$

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

Add $- 9 c$ and $c$ .

$a = - b - c$

$2 = - 12 b - 15 c$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$2 = - 12 b - 15 c$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$2 = - 12 b - 15 c$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$2 = - 12 b - 15 c$

$- 7 = - 6 b - 8 c$

Solve for $b$ in the second equation.

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Rewrite the equation as $- 12 b - 15 c = 2$ .

$a = - b - c$

$- 12 b - 15 c = 2$

$- 7 = - 6 b - 8 c$

Add $15 c$ to both sides of the equation.

$a = - b - c$

$- 12 b = 2 + 15 c$

$- 7 = - 6 b - 8 c$

Divide each term by $- 12$ and simplify.

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Divide each term in $- 12 b = 2 + 15 c$ by $- 12$ .

$a = - b - c$

$\frac{- 12 b}{- 12} = \frac{2}{- 12} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Cancel the common factor of $- 12$ .

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Cancel the common factor.

$a = - b - c$

$\frac{- 12 b}{- 12} = \frac{2}{- 12} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Divide $b$ by $1$ .

$a = - b - c$

$b = \frac{2}{- 12} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = \frac{2}{- 12} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Simplify each term.

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Cancel the common factor of $2$ and $- 12$ .

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Factor $2$ out of $2$ .

$a = - b - c$

$b = \frac{2 (1)}{- 12} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Cancel the common factors.

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Factor $2$ out of $- 12$ .

$a = - b - c$

$b = \frac{2 \cdot 1}{2 \cdot - 6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Cancel the common factor.

$a = - b - c$

$b = \frac{2 \cdot 1}{2 \cdot - 6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Rewrite the expression.

$a = - b - c$

$b = \frac{1}{- 6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = \frac{1}{- 6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = \frac{1}{- 6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Move the negative in front of the fraction.

$a = - b - c$

$b = - \frac{1}{6} + \frac{15 c}{- 12}$

$- 7 = - 6 b - 8 c$

Cancel the common factor of $15$ and $- 12$ .

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Factor $3$ out of $15 c$ .

$a = - b - c$

$b = - \frac{1}{6} + \frac{3 (5 c)}{- 12}$

$- 7 = - 6 b - 8 c$

Cancel the common factors.

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Factor $3$ out of $- 12$ .

$a = - b - c$

$b = - \frac{1}{6} + \frac{3 (5 c)}{3 (- 4)}$

$- 7 = - 6 b - 8 c$

Cancel the common factor.

$a = - b - c$

$b = - \frac{1}{6} + \frac{3 (5 c)}{3 \cdot - 4}$

$- 7 = - 6 b - 8 c$

Rewrite the expression.

$a = - b - c$

$b = - \frac{1}{6} + \frac{5 c}{- 4}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = - \frac{1}{6} + \frac{5 c}{- 4}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = - \frac{1}{6} + \frac{5 c}{- 4}$

$- 7 = - 6 b - 8 c$

Move the negative in front of the fraction.

$a = - b - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 b - 8 c$

$a = - b - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 b - 8 c$

Replace all occurrences of $b$ with $- \frac{1}{6} - \frac{5 c}{4}$ in each equation.

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Replace all occurrences of $b$ in $a = - b - c$ with $- \frac{1}{6} - \frac{5 c}{4}$ .

$a = - (- \frac{1}{6} - \frac{5 c}{4}) - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 b - 8 c$

Replace all occurrences of $b$ in $- 7 = - 6 b - 8 c$ with $- \frac{1}{6} - \frac{5 c}{4}$ .

$a = - (- \frac{1}{6} - \frac{5 c}{4}) - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = - (- \frac{1}{6} - \frac{5 c}{4}) - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Simplify each equation.

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Simplify $- (- \frac{1}{6} - \frac{5 c}{4}) - c$ .

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Simplify each term.

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Apply the distributive property.

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $- - \frac{1}{6}$ .

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Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{1}{6}) + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $\frac{1}{6}$ by $1$ .

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $- - \frac{5 c}{4}$ .

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Multiply $- 1$ by $- 1$ .

$a = \frac{1}{6} + 1 (\frac{5 c}{4}) - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $\frac{5 c}{4}$ by $1$ .

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{5 c}{4} - c$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

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

$a = \frac{1}{6} + \frac{5 c}{4} - c \cdot \frac{4}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Simplify terms.

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

$a = \frac{1}{6} + \frac{5 c}{4} + \frac{- c \cdot 4}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Combine the numerators over the common denominator.

$a = \frac{1}{6} + \frac{5 c - c \cdot 4}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{5 c - c \cdot 4}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Simplify each term.

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Simplify the numerator.

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Factor $c$ out of $5 c - c \cdot 4$ .

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Factor $c$ out of $5 c$ .

$a = \frac{1}{6} + \frac{c \cdot 5 - c \cdot 4}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Factor $c$ out of $- c \cdot 4$ .

$a = \frac{1}{6} + \frac{c \cdot 5 + c (- 1 \cdot 4)}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Factor $c$ out of $c \cdot 5 + c (- 1 \cdot 4)$ .

$a = \frac{1}{6} + \frac{c (5 - 1 \cdot 4)}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{c (5 - 1 \cdot 4)}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $- 1$ by $4$ .

$a = \frac{1}{6} + \frac{c (5 - 4)}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Subtract $4$ from $5$ .

$a = \frac{1}{6} + \frac{c \cdot 1}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{c \cdot 1}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Multiply $c$ by $1$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$

Simplify $- 6 (- \frac{1}{6} - \frac{5 c}{4}) - 8 c$ .

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Simplify each term.

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Apply the distributive property.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (- \frac{1}{6}) - 6 (- \frac{5 c}{4}) - 8 c$

Cancel the common factor of $6$ .

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Move the leading negative in $- \frac{1}{6}$ into the numerator.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 6 (\frac{- 1}{6}) - 6 (- \frac{5 c}{4}) - 8 c$

Factor $6$ out of $- 6$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 6 (- 1) (\frac{- 1}{6}) - 6 (- \frac{5 c}{4}) - 8 c$

Cancel the common factor.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 6 \cdot (- 1 (\frac{- 1}{6})) - 6 (- \frac{5 c}{4}) - 8 c$

Rewrite the expression.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 1 \cdot - 1 - 6 (- \frac{5 c}{4}) - 8 c$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = - 1 \cdot - 1 - 6 (- \frac{5 c}{4}) - 8 c$

Multiply $- 1$ by $- 1$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - 6 (- \frac{5 c}{4}) - 8 c$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{5 c}{4}$ into the numerator.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - 6 \frac{- 5 c}{4} - 8 c$

Factor $2$ out of $- 6$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + 2 (- 3) (\frac{- 5 c}{4}) - 8 c$

Factor $2$ out of $4$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + 2 \cdot (- 3 \frac{- 5 c}{2 \cdot 2}) - 8 c$

Cancel the common factor.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + 2 \cdot (- 3 \frac{- 5 c}{2 \cdot 2}) - 8 c$

Rewrite the expression.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - 3 \frac{- 5 c}{2} - 8 c$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - 3 \frac{- 5 c}{2} - 8 c$

Combine $- 3$ and $\frac{- 5 c}{2}$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{- 3 (- 5 c)}{2} - 8 c$

Multiply $- 5$ by $- 3$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c}{2} - 8 c$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c}{2} - 8 c$

To write $- 8 c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c}{2} - 8 c \cdot \frac{2}{2}$

Simplify terms.

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Combine $- 8 c$ and $\frac{2}{2}$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c}{2} + \frac{- 8 c \cdot 2}{2}$

Combine the numerators over the common denominator.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c - 8 c \cdot 2}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{15 c - 8 c \cdot 2}{2}$

Simplify each term.

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Simplify the numerator.

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Factor $c$ out of $15 c - 8 c \cdot 2$ .

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Factor $c$ out of $15 c$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c \cdot 15 - 8 c \cdot 2}{2}$

Factor $c$ out of $- 8 c \cdot 2$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c \cdot 15 + c (- 8 \cdot 2)}{2}$

Factor $c$ out of $c \cdot 15 + c (- 8 \cdot 2)$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c (15 - 8 \cdot 2)}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c (15 - 8 \cdot 2)}{2}$

Multiply $- 8$ by $2$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c (15 - 16)}{2}$

Subtract $16$ from $15$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c \cdot - 1}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{c \cdot - 1}{2}$

Move $- 1$ to the left of $c$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 + \frac{- 1 \cdot c}{2}$

Move the negative in front of the fraction.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - \frac{c}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - \frac{c}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - \frac{c}{2}$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 7 = 1 - \frac{c}{2}$

Solve for $c$ in the third equation.

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Rewrite the equation as $1 - \frac{c}{2} = - 7$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$1 - \frac{c}{2} = - 7$

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

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Subtract $1$ from both sides of the equation.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- \frac{c}{2} = - 7 - 1$

Subtract $1$ from $- 7$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- \frac{c}{2} = - 8$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- \frac{c}{2} = - 8$

Multiply both sides of the equation by $- 2$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 2 \cdot (- \frac{c}{2}) = - 2 \cdot - 8$

Simplify both sides of the equation.

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Simplify $- 2 \cdot (- \frac{c}{2})$ .

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Cancel the common factor of $2$ .

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

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 2 \cdot \frac{- c}{2} = - 2 \cdot - 8$

Factor $2$ out of $- 2$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$2 (- 1) \cdot \frac{- c}{2} = - 2 \cdot - 8$

Cancel the common factor.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$2 \cdot - 1 \cdot \frac{- c}{2} = - 2 \cdot - 8$

Rewrite the expression.

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 1 \cdot (- c) = - 2 \cdot - 8$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$- 1 \cdot (- c) = - 2 \cdot - 8$

Multiply.

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Multiply $- 1$ by $- 1$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$1 \cdot c = - 2 \cdot - 8$

Multiply $c$ by $1$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = - 2 \cdot - 8$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = - 2 \cdot - 8$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = - 2 \cdot - 8$

Multiply $- 2$ by $- 8$ .

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = 16$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = 16$

$a = \frac{1}{6} + \frac{c}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = 16$

Replace all occurrences of $c$ with $16$ in each equation.

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Replace all occurrences of $c$ in $a = \frac{1}{6} + \frac{c}{4}$ with $16$ .

$a = \frac{1}{6} + \frac{16}{4}$

$b = - \frac{1}{6} - \frac{5 c}{4}$

$c = 16$

Replace all occurrences of $c$ in $b = - \frac{1}{6} - \frac{5 c}{4}$ with $16$ .

$a = \frac{1}{6} + \frac{16}{4}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

$a = \frac{1}{6} + \frac{16}{4}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Simplify.

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Simplify $\frac{1}{6} + \frac{16}{4}$ .

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Divide $16$ by $4$ .

$a = \frac{1}{6} + 4$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

To write $4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$a = \frac{1}{6} + 4 \cdot \frac{6}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Combine $4$ and $\frac{6}{6}$ .

$a = \frac{1}{6} + \frac{4 \cdot 6}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Combine the numerators over the common denominator.

$a = \frac{1 + 4 \cdot 6}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Simplify the numerator.

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Multiply $4$ by $6$ .

$a = \frac{1 + 24}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Add $1$ and $24$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{5 (16)}{4}$

$c = 16$

Simplify $- \frac{1}{6} - \frac{5 (16)}{4}$ .

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Simplify each term.

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Cancel the common factor of $16$ and $4$ .

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Factor $4$ out of $5 (16)$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{4 (5 \cdot (4))}{4}$

$c = 16$

Cancel the common factors.

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Factor $4$ out of $4$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{4 (5 \cdot (4))}{4 (1)}$

$c = 16$

Cancel the common factor.

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{4 (5 \cdot (4))}{4 \cdot 1}$

$c = 16$

Rewrite the expression.

$a = \frac{25}{6}$

$b = - \frac{1}{6} - \frac{5 \cdot (4)}{1}$

$c = 16$

Divide $5 \cdot (4)$ by $1$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - (5 \cdot (4))$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{1}{6} - (5 \cdot (4))$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{1}{6} - (5 \cdot (4))$

$c = 16$

Multiply $5$ by $4$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - 1 \cdot 20$

$c = 16$

Multiply $- 1$ by $20$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - 20$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{1}{6} - 20$

$c = 16$

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} - 20 \cdot \frac{6}{6}$

$c = 16$

Combine $- 20$ and $\frac{6}{6}$ .

$a = \frac{25}{6}$

$b = - \frac{1}{6} + \frac{- 20 \cdot 6}{6}$

$c = 16$

Combine the numerators over the common denominator.

$a = \frac{25}{6}$

$b = \frac{- 1 - 20 \cdot 6}{6}$

$c = 16$

Simplify the numerator.

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Multiply $- 20$ by $6$ .

$a = \frac{25}{6}$

$b = \frac{- 1 - 120}{6}$

$c = 16$

Subtract $120$ from $- 1$ .

$a = \frac{25}{6}$

$b = \frac{- 121}{6}$

$c = 16$

$a = \frac{25}{6}$

$b = \frac{- 121}{6}$

$c = 16$

Move the negative in front of the fraction.

$a = \frac{25}{6}$

$b = - \frac{121}{6}$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{121}{6}$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{121}{6}$

$c = 16$

$a = \frac{25}{6}$

$b = - \frac{121}{6}$

$c = 16$

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

$y = \frac{25 x^{2}}{6} - \frac{121 x}{6} + 16$

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