Simplify (x^2-14x+49)/(x^2-49)

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

Step 1

Factor using the perfect square rule.

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

$\frac{x^{2} - 14 x + 7^{2}}{x^{2} - 49}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{{(x - 7)}^{2}}{x^{2} - 49}$

Step 2

Simplify the denominator.

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

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

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 $b = 7$ .

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

Step 3

Cancel the common factor of ${(x - 7)}^{2}$ and $x - 7$ .

Factor $x - 7$ out of ${(x - 7)}^{2}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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