Algebra Examples

$\frac{\sqrt{x^{2} + 9} - 5}{x + 4}$

Step 1

Multiply to rationalize the numerator.

$\frac{(\sqrt{x^{2} + 9} - 5) (\sqrt{x^{2} + 9} + 5)}{(x + 4) (\sqrt{x^{2} + 9} + 5)}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$\frac{{\sqrt{x^{2} + 9}}^{2} + \sqrt{x^{2} + 9} \cdot 5 + \sqrt{x^{2} + 9} \cdot - 5 - 25}{(x + 4) (\sqrt{x^{2} + 9} + 5)}$

Step 2.2

Simplify.

$\frac{x^{2} - 16}{(x + 4) (\sqrt{x^{2} + 9} + 5)}$

Step 3

Expand $(x + 4) (\sqrt{x^{2} + 9} + 5)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{x^{2} - 16}{x (\sqrt{x^{2} + 9} + 5) + 4 (\sqrt{x^{2} + 9} + 5)}$

Step 3.2

Apply the distributive property.

$\frac{x^{2} - 16}{x \sqrt{x^{2} + 9} + x \cdot 5 + 4 (\sqrt{x^{2} + 9} + 5)}$

Step 3.3

Apply the distributive property.

$\frac{x^{2} - 16}{x \sqrt{x^{2} + 9} + x \cdot 5 + 4 \sqrt{x^{2} + 9} + 4 \cdot 5}$

Step 4

Simplify each term.

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Step 4.1

Move $5$ to the left of $x$ .

$\frac{x^{2} - 16}{x \sqrt{x^{2} + 9} + 5 \cdot x + 4 \sqrt{x^{2} + 9} + 4 \cdot 5}$

Step 4.2

Multiply $4$ by $5$ .

$\frac{x^{2} - 16}{x \sqrt{x^{2} + 9} + 5 x + 4 \sqrt{x^{2} + 9} + 20}$