Algebra Examples

$y = \frac{1}{8} x^{2} + 4 x + 20$

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.

$\frac{1}{8} x^{2} + 4 x + 20 = y$

Step 2

Subtract $20$ from both sides of the equation.

$\frac{1}{8} x^{2} + 4 x = y - 20$

Step 3

Complete the square.

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Step 3.1

Combine $\frac{1}{8}$ and $x^{2}$ .

$\frac{x^{2}}{8} + 4 x = y - 20$

Step 3.2

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

$a = \frac{1}{8}$

$b = 4$

$c = 0 = y - 20$

Step 3.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 20$

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 $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 (\frac{1}{8})}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Cancel the common factor of $4$ and $2$ .

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Step 3.4.2.1.1

Factor $2$ out of $4$ .

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

Step 3.4.2.1.2

Cancel the common factors.

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Step 3.4.2.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.2.2

Rewrite the expression.

$d = \frac{2}{\frac{1}{8}}$

Step 3.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 2 \cdot 8$

Step 3.4.2.3

Multiply $2$ by $8$ .

$d = 16$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{4^{2}}{4 (\frac{1}{8})}$

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 3.5.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 (\frac{1}{8})}$

Step 3.5.2.1.1.2

Cancel the common factors.

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Step 3.5.2.1.1.2.1

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 (\frac{1}{8})}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

$e = 0 - \frac{4}{\frac{1}{8}}$

Step 3.5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (4 \cdot 8)$

Step 3.5.2.1.3

Multiply $- (4 \cdot 8)$ .

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Step 3.5.2.1.3.1

Multiply $4$ by $8$ .

$e = 0 - 1 \cdot 32$

Step 3.5.2.1.3.2

Multiply $- 1$ by $32$ .

$e = 0 - 32$

Step 3.5.2.2

Subtract $32$ from $0$ .

$e = - 32$

Step 3.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x + 16)}^{2} - 32$ .

$\frac{1}{8} \cdot {(x + 16)}^{2} - 32 = y - 20$

Step 4

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

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Step 4.1

Add $32$ to both sides of the equation.

$\frac{1}{8} \cdot {(x + 16)}^{2} = y - 20 + 32$

Step 4.2

Add $- 20$ and $32$ .

$\frac{1}{8} \cdot {(x + 16)}^{2} = y + 12$

Step 5

Multiply each term in $\frac{1}{8} \cdot {(x + 16)}^{2} = y + 12$ by $8$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{1}{8} \cdot {(x + 16)}^{2} = y + 12$ by $8$ .

$\frac{1}{8} \cdot {(x + 16)}^{2} \cdot 8 = y \cdot 8 + 12 \cdot 8$

Step 5.2

Simplify the left side.

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Step 5.2.1

Combine $\frac{1}{8}$ and ${(x + 16)}^{2}$ .

$\frac{{(x + 16)}^{2}}{8} \cdot 8 = y \cdot 8 + 12 \cdot 8$

Step 5.2.2

Cancel the common factor of $8$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{{(x + 16)}^{2}}{8} \cdot 8 = y \cdot 8 + 12 \cdot 8$

Step 5.2.2.2

Rewrite the expression.

${(x + 16)}^{2} = y \cdot 8 + 12 \cdot 8$

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 $8$ to the left of $y$ .

${(x + 16)}^{2} = 8 \cdot y + 12 \cdot 8$

Step 5.3.1.2

Multiply $12$ by $8$ .

${(x + 16)}^{2} = 8 y + 96$

Step 6

Factor $8$ out of $8 y + 96$ .

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Step 6.1

Factor $8$ out of $8 y$ .

${(x + 16)}^{2} = 8 (y) + 96$

Step 6.2

Factor $8$ out of $96$ .

${(x + 16)}^{2} = 8 y + 8 \cdot 12$

Step 6.3

Factor $8$ out of $8 y + 8 \cdot 12$ .

${(x + 16)}^{2} = 8 (y + 12)$