Simplify (x^2+4x-21)/(x^2-49)-8/(2x-14)

$\frac{x^{2} + 4 x - 21}{x^{2} - 49} - \frac{8}{2 x - 14}$

Simplify each term.

Factor $x^{2} + 4 x - 21$ 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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Write the factored form using these integers.

$\frac{(x - 3) (x + 7)}{x^{2} - 49} - \frac{8}{2 x - 14}$

Simplify the denominator.

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

$\frac{(x - 3) (x + 7)}{x^{2} - 7^{2}} - \frac{8}{2 x - 14}$

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 $b = 7$ .

$\frac{(x - 3) (x + 7)}{(x + 7) (x - 7)} - \frac{8}{2 x - 14}$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{(x - 3) (x + 7)}{(x + 7) (x - 7)} - \frac{8}{2 x - 14}$

Rewrite the expression.

$\frac{x - 3}{x - 7} - \frac{8}{2 x - 14}$

Cancel the common factor of $8$ and $2 x - 14$ .

Factor $2$ out of $8$ .

$\frac{x - 3}{x - 7} - \frac{2 \cdot 4}{2 x - 14}$

Cancel the common factors.

Factor $2$ out of $2 x$ .

$\frac{x - 3}{x - 7} - \frac{2 \cdot 4}{2 (x) - 14}$

Factor $2$ out of $- 14$ .

$\frac{x - 3}{x - 7} - \frac{2 \cdot 4}{2 (x) + 2 (- 7)}$

Factor $2$ out of $2 (x) + 2 (- 7)$ .

$\frac{x - 3}{x - 7} - \frac{2 \cdot 4}{2 (x - 7)}$

Cancel the common factor.

$\frac{x - 3}{x - 7} - \frac{2 \cdot 4}{2 (x - 7)}$

Rewrite the expression.

$\frac{x - 3}{x - 7} - \frac{4}{x - 7}$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{x - 3 - 4}{x - 7}$

Subtract $4$ from $- 3$ .

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

Cancel the common factor of $x - 7$ .

Cancel the common factor.

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

Rewrite the expression.

$1$

Simplify (x^2+4x-21)/(x^2-49)-8/(2x-14)

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