Algebra Examples

$\frac{x}{x^{2} - 64} - \frac{7}{4 x^{2} + 32 x}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $64$ as $8^{2}$ .

$\frac{x}{x^{2} - 8^{2}} - \frac{7}{4 x^{2} + 32 x}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 8$ .

$\frac{x}{(x + 8) (x - 8)} - \frac{7}{4 x^{2} + 32 x}$

Step 1.2

Factor $4 x$ out of $4 x^{2} + 32 x$ .

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Step 1.2.1

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

$\frac{x}{(x + 8) (x - 8)} - \frac{7}{4 x (x) + 32 x}$

Step 1.2.2

Factor $4 x$ out of $32 x$ .

$\frac{x}{(x + 8) (x - 8)} - \frac{7}{4 x (x) + 4 x (8)}$

Step 1.2.3

Factor $4 x$ out of $4 x (x) + 4 x (8)$ .

$\frac{x}{(x + 8) (x - 8)} - \frac{7}{4 x (x + 8)}$

Step 2

To write $\frac{x}{(x + 8) (x - 8)}$ as a fraction with a common denominator, multiply by $\frac{4 x}{4 x}$ .

$\frac{x}{(x + 8) (x - 8)} \cdot \frac{4 x}{4 x} - \frac{7}{4 x (x + 8)}$

Step 3

To write $- \frac{7}{4 x (x + 8)}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{x}{(x + 8) (x - 8)} \cdot \frac{4 x}{4 x} - \frac{7}{4 x (x + 8)} \cdot \frac{x - 8}{x - 8}$

Step 4

Write each expression with a common denominator of $(x + 8) (x - 8) \cdot 4 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{(x + 8) (x - 8)}$ by $\frac{4 x}{4 x}$ .

$\frac{x (4 x)}{(x + 8) (x - 8) (4 x)} - \frac{7}{4 x (x + 8)} \cdot \frac{x - 8}{x - 8}$

Step 4.2

Multiply $\frac{7}{4 x (x + 8)}$ by $\frac{x - 8}{x - 8}$ .

$\frac{x (4 x)}{(x + 8) (x - 8) (4 x)} - \frac{7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 4.3

Reorder the factors of $(x + 8) (x - 8) (4 x)$ .

$\frac{x (4 x)}{4 x (x + 8) (x - 8)} - \frac{7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (4 x) - 7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite using the commutative property of multiplication.

$\frac{4 x \cdot x - 7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{4 (x \cdot x) - 7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} - 7 (x - 8)}{4 x (x + 8) (x - 8)}$

Step 6.3

Apply the distributive property.

$\frac{4 x^{2} - 7 x - 7 \cdot - 8}{4 x (x + 8) (x - 8)}$

Step 6.4

Multiply $- 7$ by $- 8$ .

$\frac{4 x^{2} - 7 x + 56}{4 x (x + 8) (x - 8)}$