Algebra Examples

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

Step 1

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

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

Step 2

Complete the square for $\frac{x^{2}}{2} - 4 x$ .

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Step 2.1

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

$a = \frac{1}{2}$

$b = - 4$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $- 4$ .

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

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.2.2

Rewrite the expression.

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

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 2 \cdot 2$

Step 2.3.2.3

Multiply $- 2$ by $2$ .

$d = - 4$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

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

Step 2.4.2.1.1.3

Raise $- 1$ to the power of $2$ .

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

Step 2.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

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

Step 2.4.2.1.1.5

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

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

Step 2.4.2.1.1.6

Cancel the common factors.

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Step 2.4.2.1.1.6.1

Cancel the common factor.

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

Step 2.4.2.1.1.6.2

Rewrite the expression.

$e = 0 - \frac{4}{\frac{1}{2}}$

Step 2.4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (4 \cdot 2)$

Step 2.4.2.1.3

Multiply $- (4 \cdot 2)$ .

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Step 2.4.2.1.3.1

Multiply $4$ by $2$ .

$e = 0 - 1 \cdot 8$

Step 2.4.2.1.3.2

Multiply $- 1$ by $8$ .

$e = 0 - 8$

Step 2.4.2.2

Subtract $8$ from $0$ .

$e = - 8$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 4)}^{2} - 8$ .

$\frac{1}{2} {(x - 4)}^{2} - 8$

Step 3

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x - 4)}^{2} - 8$