Simplify (x^2)/(x^2-36)-(x+42)/(x^2-36)

$\frac{x^{2}}{x^{2} - 36} - \frac{x + 42}{x^{2} - 36}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{x^{2} - (x + 42)}{x^{2} - 36}$

Simplify each term.

Apply the distributive property.

$\frac{x^{2} - x - 1 \cdot 42}{x^{2} - 36}$

Multiply $- 1$ by $42$ .

$\frac{x^{2} - x - 42}{x^{2} - 36}$

Factor $x^{2} - x - 42$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Write the factored form using these integers.

$\frac{(x - 7) (x + 6)}{x^{2} - 36}$

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$\frac{(x - 7) (x + 6)}{x^{2} - 6^{2}}$

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 $b = 6$ .

$\frac{(x - 7) (x + 6)}{(x + 6) (x - 6)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x - 7) (x + 6)}{(x + 6) (x - 6)}$

Rewrite the expression.

$\frac{x - 7}{x - 6}$

Simplify (x^2)/(x^2-36)-(x+42)/(x^2-36)

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