Precalculus Examples

$(\sqrt{x^{2} - 16}) + (\frac{x^{2}}{x^{2} + 1})$

Step 1

Remove parentheses.

$\sqrt{x^{2} - 16} + \frac{x^{2}}{x^{2} + 1}$

Step 2

Simplify each term.

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Step 2.1

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

$\sqrt{x^{2} - 4^{2}} + \frac{x^{2}}{x^{2} + 1}$

Step 2.2

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

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

Step 3

To write $\sqrt{(x + 4) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $\sqrt{(x + 4) (x - 4)}$ by $1$ .

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