Algebra Examples

$y = (x - 4) (x + 4)$

Step 1

Complete the square for $(x - 4) (x + 4)$ .

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Step 1.1

Simplify the expression.

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Step 1.1.1

Expand $(x - 4) (x + 4)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$x (x + 4) - 4 (x + 4)$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 4 - 4 (x + 4)$

Step 1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$

Step 1.1.2

Combine the opposite terms in $x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$ .

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Step 1.1.2.1

Reorder the factors in the terms $x \cdot 4$ and $- 4 x$ .

$x \cdot x + 4 x - 4 x - 4 \cdot 4$

Step 1.1.2.2

Subtract $4 x$ from $4 x$ .

$x \cdot x + 0 - 4 \cdot 4$

Step 1.1.2.3

Add $x \cdot x$ and $0$ .

$x \cdot x - 4 \cdot 4$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Multiply $x$ by $x$ .

$x^{2} - 4 \cdot 4$

Step 1.1.3.2

Multiply $- 4$ by $4$ .

$x^{2} - 16$

Step 1.2

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

$a = 1$

$b = 0$

$c = - 16$

Step 1.3

Consider the vertex form of a parabola.

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

Step 1.4

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

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Step 1.4.1

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

$d = \frac{0}{2 \cdot 1}$

Step 1.4.2

Cancel the common factor of $0$ and $2$ .

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Step 1.4.2.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 1}$

Step 1.4.2.2

Cancel the common factors.

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Step 1.4.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 (0)}{2 (1)}$

Step 1.4.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 1.4.2.2.3

Rewrite the expression.

$d = \frac{0}{1}$

Step 1.4.2.2.4

Divide $0$ by $1$ .

$d = 0$

Step 1.5

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

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Step 1.5.1

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

$e = - 16 - \frac{0^{2}}{4 \cdot 1}$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = - 16 - \frac{0}{4 \cdot 1}$

Step 1.5.2.1.2

Multiply $4$ by $1$ .

$e = - 16 - \frac{0}{4}$

Step 1.5.2.1.3

Divide $0$ by $4$ .

$e = - 16 - 0$

Step 1.5.2.1.4

Multiply $- 1$ by $0$ .

$e = - 16 + 0$

Step 1.5.2.2

Add $- 16$ and $0$ .

$e = - 16$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 0)}^{2} - 16$ .

${(x + 0)}^{2} - 16$

Step 2

Set $y$ equal to the new right side.

$y = {(x + 0)}^{2} - 16$