Pre-Algebra Examples

$\frac{q^{2} - 16}{q^{2} - 8 q + 16}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

$\frac{q^{2} - 4^{2}}{q^{2} - 8 q + 16}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = q$ and $b = 4$ .

$\frac{(q + 4) (q - 4)}{q^{2} - 8 q + 16}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{(q + 4) (q - 4)}{q^{2} - 8 q + 4^{2}}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 q = 2 \cdot q \cdot 4$

Step 2.3

Rewrite the polynomial.

$\frac{(q + 4) (q - 4)}{q^{2} - 2 \cdot q \cdot 4 + 4^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = q$ and $b = 4$ .

$\frac{(q + 4) (q - 4)}{{(q - 4)}^{2}}$

Step 3

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

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Step 3.1

Factor $q - 4$ out of $(q + 4) (q - 4)$ .

$\frac{(q - 4) (q + 4)}{{(q - 4)}^{2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $q - 4$ out of ${(q - 4)}^{2}$ .

$\frac{(q - 4) (q + 4)}{(q - 4) (q - 4)}$

Step 3.2.2

Cancel the common factor.

$\frac{(q - 4) (q + 4)}{(q - 4) (q - 4)}$

Step 3.2.3

Rewrite the expression.

$\frac{q + 4}{q - 4}$