Algebra Examples

$y = - \frac{1}{6} x^{2} + 7 x - 80$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{1}{6} x^{2} + 7 x - 80 = y$

Step 2

Add $80$ to both sides of the equation.

$- \frac{1}{6} x^{2} + 7 x = y + 80$

Step 3

Complete the square.

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Step 3.1

Combine $x^{2}$ and $\frac{1}{6}$ .

$- \frac{x^{2}}{6} + 7 x = y + 80$

Step 3.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{6}$

$b = 7$

$c = 0 = y + 80$

Step 3.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 80$

Step 3.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{7}{2 (- \frac{1}{6})}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify the denominator.

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Step 3.4.2.1.1

Multiply $- 1$ by $2$ .

$d = \frac{7}{- 2 (\frac{1}{6})}$

Step 3.4.2.1.2

Combine $- 2$ and $\frac{1}{6}$ .

$d = \frac{7}{\frac{- 2}{6}}$

Step 3.4.2.2

Reduce the expression by cancelling the common factors.

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Step 3.4.2.2.1

Cancel the common factor of $- 2$ and $6$ .

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Step 3.4.2.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{7}{\frac{2 (- 1)}{6}}$

Step 3.4.2.2.1.2

Cancel the common factors.

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Step 3.4.2.2.1.2.1

Factor $2$ out of $6$ .

$d = \frac{7}{\frac{2 \cdot - 1}{2 \cdot 3}}$

Step 3.4.2.2.1.2.2

Cancel the common factor.

$d = \frac{7}{\frac{2 \cdot - 1}{2 \cdot 3}}$

Step 3.4.2.2.1.2.3

Rewrite the expression.

$d = \frac{7}{\frac{- 1}{3}}$

Step 3.4.2.2.2

Move the negative in front of the fraction.

$d = \frac{7}{- \frac{1}{3}}$

Step 3.4.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 7 (- 1 \cdot 3)$

Step 3.4.2.4

Multiply $7 (- 1 \cdot 3)$ .

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Step 3.4.2.4.1

Multiply $- 1$ by $3$ .

$d = 7 \cdot - 3$

Step 3.4.2.4.2

Multiply $7$ by $- 3$ .

$d = - 21$

Step 3.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{7^{2}}{4 (- \frac{1}{6})}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Raise $7$ to the power of $2$ .

$e = 0 - \frac{49}{4 (- \frac{1}{6})}$

Step 3.5.2.1.2

Simplify the denominator.

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Step 3.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{49}{- 4 (\frac{1}{6})}$

Step 3.5.2.1.2.2

Combine $- 4$ and $\frac{1}{6}$ .

$e = 0 - \frac{49}{\frac{- 4}{6}}$

Step 3.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 3.5.2.1.3.1

Cancel the common factor of $- 4$ and $6$ .

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Step 3.5.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = 0 - \frac{49}{\frac{2 (- 2)}{6}}$

Step 3.5.2.1.3.1.2

Cancel the common factors.

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Step 3.5.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = 0 - \frac{49}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 3.5.2.1.3.1.2.2

Cancel the common factor.

$e = 0 - \frac{49}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 3.5.2.1.3.1.2.3

Rewrite the expression.

$e = 0 - \frac{49}{\frac{- 2}{3}}$

Step 3.5.2.1.3.2

Move the negative in front of the fraction.

$e = 0 - \frac{49}{- \frac{2}{3}}$

Step 3.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (49 (- \frac{3}{2}))$

Step 3.5.2.1.5

Multiply $49 (- \frac{3}{2})$ .

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Step 3.5.2.1.5.1

Multiply $- 1$ by $49$ .

$e = 0 - (- 49 (\frac{3}{2}))$

Step 3.5.2.1.5.2

Combine $- 49$ and $\frac{3}{2}$ .

$e = 0 - \frac{- 49 \cdot 3}{2}$

Step 3.5.2.1.5.3

Multiply $- 49$ by $3$ .

$e = 0 - \frac{- 147}{2}$

Step 3.5.2.1.6

Move the negative in front of the fraction.

$e = 0 - - \frac{147}{2}$

Step 3.5.2.1.7

Multiply $- - \frac{147}{2}$ .

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Step 3.5.2.1.7.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{147}{2})$

Step 3.5.2.1.7.2

Multiply $\frac{147}{2}$ by $1$ .

$e = 0 + \frac{147}{2}$

Step 3.5.2.2

Add $0$ and $\frac{147}{2}$ .

$e = \frac{147}{2}$

Step 3.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{6} {(x - 21)}^{2} + \frac{147}{2}$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} + \frac{147}{2} = y + 80$

Step 4

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

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Step 4.1

Subtract $\frac{147}{2}$ from both sides of the equation.

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + 80 - \frac{147}{2}$

Step 4.2

To write $80$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + 80 \cdot \frac{2}{2} - \frac{147}{2}$

Step 4.3

Combine $80$ and $\frac{2}{2}$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{80 \cdot 2}{2} - \frac{147}{2}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{80 \cdot 2 - 147}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $80$ by $2$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{160 - 147}{2}$

Step 4.5.2

Subtract $147$ from $160$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{13}{2}$

Step 5

Multiply each term in $- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{13}{2}$ by $- 6$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{1}{6} \cdot {(x - 21)}^{2} = y + \frac{13}{2}$ by $- 6$ .

$- \frac{1}{6} \cdot {(x - 21)}^{2} \cdot - 6 = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2

Simplify the left side.

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Step 5.2.1

Combine ${(x - 21)}^{2}$ and $\frac{1}{6}$ .

$- \frac{{(x - 21)}^{2}}{6} \cdot - 6 = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.2

Cancel the common factor of $6$ .

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Step 5.2.2.1

Move the leading negative in $- \frac{{(x - 21)}^{2}}{6}$ into the numerator.

$\frac{- {(x - 21)}^{2}}{6} \cdot - 6 = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.2.2

Factor $6$ out of $- 6$ .

$\frac{- {(x - 21)}^{2}}{6} \cdot (6 (- 1)) = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.2.3

Cancel the common factor.

$\frac{- {(x - 21)}^{2}}{6} \cdot (6 \cdot - 1) = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.2.4

Rewrite the expression.

$- {(x - 21)}^{2} \cdot - 1 = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.3

Multiply.

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Step 5.2.3.1

Multiply $- 1$ by $- 1$ .

$1 {(x - 21)}^{2} = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.2.3.2

Multiply ${(x - 21)}^{2}$ by $1$ .

${(x - 21)}^{2} = y \cdot - 6 + \frac{13}{2} \cdot - 6$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

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

${(x - 21)}^{2} = - 6 \cdot y + \frac{13}{2} \cdot - 6$

Step 5.3.1.2

Cancel the common factor of $2$ .

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Step 5.3.1.2.1

Factor $2$ out of $- 6$ .

${(x - 21)}^{2} = - 6 y + \frac{13}{2} \cdot (2 (- 3))$

Step 5.3.1.2.2

Cancel the common factor.

${(x - 21)}^{2} = - 6 y + \frac{13}{2} \cdot (2 \cdot - 3)$

Step 5.3.1.2.3

Rewrite the expression.

${(x - 21)}^{2} = - 6 y + 13 \cdot - 3$

Step 5.3.1.3

Multiply $13$ by $- 3$ .

${(x - 21)}^{2} = - 6 y - 39$

Step 6

Factor $3$ out of $- 6 y - 39$ .

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Step 6.1

Factor $3$ out of $- 6 y$ .

${(x - 21)}^{2} = 3 (- 2 y) - 39$

Step 6.2

Factor $3$ out of $- 39$ .

${(x - 21)}^{2} = 3 (- 2 y) + 3 (- 13)$

Step 6.3

Factor $3$ out of $3 (- 2 y) + 3 (- 13)$ .

${(x - 21)}^{2} = 3 (- 2 y - 13)$