Algebra Examples

$\frac{x^{2}}{x^{2} - 16} + \frac{8 (x - 2)}{16 - x^{2}}$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2}}{x^{2} - 4^{2}} + \frac{8 (x - 2)}{16 - x^{2}}$

Step 1.1.1.2

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 $b = 4$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{16 - x^{2}}$

Step 1.1.2

Simplify the denominator.

Tap for more steps...

Step 1.1.2.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

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

Step 1.2

Simplify terms.

Tap for more steps...

Step 1.2.1

Reorder terms.

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

Step 1.2.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4) - x)}$

Step 1.2.3

Factor $- 1$ out of $- x$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4) - (x))}$

Step 1.2.4

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$\frac{x^{2}}{(x + 4) (x - 4)} + \frac{8 (x - 2)}{(x + 4) (- 1 (- 4 + x))}$

Step 1.2.5

Simplify the expression.

Tap for more steps...

Step 1.2.5.1

Move a negative from the denominator of $\frac{8 (x - 2)}{(x + 4) (- 1 (- 4 + x))}$ to the numerator.

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

Step 1.2.5.2

Reorder terms.

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

Step 1.2.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Multiply $- 1$ by $8$ .

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Multiply $- 8$ by $- 2$ .

$\frac{x^{2} - 8 x + 16}{(x + 4) (x - 4)}$

Step 2.4

Factor using the perfect square rule.

Tap for more steps...

Step 2.4.1

Rewrite $16$ as $4^{2}$ .

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

Step 2.4.2

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

$8 x = 2 \cdot x \cdot 4$

Step 2.4.3

Rewrite the polynomial.

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

Step 2.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 3

Cancel the common factor of ${(x - 4)}^{2}$ and $x - 4$ .

Tap for more steps...

Step 3.1

Factor $x - 4$ out of ${(x - 4)}^{2}$ .

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\frac{x - 4}{x + 4}$