Algebra Examples

$\frac{3}{x + 7} - \frac{42}{x^{2} - 49}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{3}{x + 7} - \frac{42}{x^{2} - 7^{2}}$

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 $a = x$ and $b = 7$ .

$\frac{3}{x + 7} - \frac{42}{(x + 7) (x - 7)}$

Step 2

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

$\frac{3}{x + 7} \cdot \frac{x - 7}{x - 7} - \frac{42}{(x + 7) (x - 7)}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{3 (x - 7)}{(x + 7) (x - 7)} - \frac{42}{(x + 7) (x - 7)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 (x - 7) - 42}{(x + 7) (x - 7)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3$ out of $3 (x - 7) - 42$ .

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Step 4.1.1

Factor $3$ out of $- 42$ .

$\frac{3 (x - 7) + 3 \cdot - 14}{(x + 7) (x - 7)}$

Step 4.1.2

Factor $3$ out of $3 (x - 7) + 3 \cdot - 14$ .

$\frac{3 (x - 7 - 14)}{(x + 7) (x - 7)}$

Step 4.2

Subtract $14$ from $- 7$ .

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