Algebra Examples

$\frac{x^{4}}{9 x^{2} - 16} - \frac{7}{12 x - 16}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{x^{4}}{{(3 x)}^{2} - 16} - \frac{7}{12 x - 16}$

Step 1.1.2

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

$\frac{x^{4}}{{(3 x)}^{2} - 4^{2}} - \frac{7}{12 x - 16}$

Step 1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 4$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} - \frac{7}{12 x - 16}$

Step 1.2

Factor $4$ out of $12 x - 16$ .

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Step 1.2.1

Factor $4$ out of $12 x$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} - \frac{7}{4 (3 x) - 16}$

Step 1.2.2

Factor $4$ out of $- 16$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} - \frac{7}{4 (3 x) + 4 (- 4)}$

Step 1.2.3

Factor $4$ out of $4 (3 x) + 4 (- 4)$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} - \frac{7}{4 (3 x - 4)}$

Step 2

To write $\frac{x^{4}}{(3 x + 4) (3 x - 4)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} \cdot \frac{4}{4} - \frac{7}{4 (3 x - 4)}$

Step 3

To write $- \frac{7}{4 (3 x - 4)}$ as a fraction with a common denominator, multiply by $\frac{3 x + 4}{3 x + 4}$ .

$\frac{x^{4}}{(3 x + 4) (3 x - 4)} \cdot \frac{4}{4} - \frac{7}{4 (3 x - 4)} \cdot \frac{3 x + 4}{3 x + 4}$

Step 4

Write each expression with a common denominator of $(3 x + 4) (3 x - 4) \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x^{4}}{(3 x + 4) (3 x - 4)}$ by $\frac{4}{4}$ .

$\frac{x^{4} \cdot 4}{(3 x + 4) (3 x - 4) \cdot 4} - \frac{7}{4 (3 x - 4)} \cdot \frac{3 x + 4}{3 x + 4}$

Step 4.2

Multiply $\frac{7}{4 (3 x - 4)}$ by $\frac{3 x + 4}{3 x + 4}$ .

$\frac{x^{4} \cdot 4}{(3 x + 4) (3 x - 4) \cdot 4} - \frac{7 (3 x + 4)}{4 (3 x - 4) (3 x + 4)}$

Step 4.3

Reorder the factors of $(3 x + 4) (3 x - 4) \cdot 4$ .

$\frac{x^{4} \cdot 4}{4 (3 x + 4) (3 x - 4)} - \frac{7 (3 x + 4)}{4 (3 x - 4) (3 x + 4)}$

Step 4.4

Reorder the factors of $4 (3 x - 4) (3 x + 4)$ .

$\frac{x^{4} \cdot 4}{4 (3 x + 4) (3 x - 4)} - \frac{7 (3 x + 4)}{4 (3 x + 4) (3 x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x^{4} \cdot 4 - 7 (3 x + 4)}{4 (3 x + 4) (3 x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Move $4$ to the left of $x^{4}$ .

$\frac{4 \cdot x^{4} - 7 (3 x + 4)}{4 (3 x + 4) (3 x - 4)}$

Step 6.2

Apply the distributive property.

$\frac{4 x^{4} - 7 (3 x) - 7 \cdot 4}{4 (3 x + 4) (3 x - 4)}$

Step 6.3

Multiply $3$ by $- 7$ .

$\frac{4 x^{4} - 21 x - 7 \cdot 4}{4 (3 x + 4) (3 x - 4)}$

Step 6.4

Multiply $- 7$ by $4$ .

$\frac{4 x^{4} - 21 x - 28}{4 (3 x + 4) (3 x - 4)}$