Algebra Examples

$\frac{4 - x^{2}}{3 - \sqrt{x^{2} + 5}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{2^{2} - x^{2}}{3 - \sqrt{x^{2} + 5}}$

Step 1.2

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

$\frac{(2 + x) (2 - x)}{3 - \sqrt{x^{2} + 5}}$

Step 2

Multiply $\frac{(2 + x) (2 - x)}{3 - \sqrt{x^{2} + 5}}$ by $\frac{3 + \sqrt{x^{2} + 5}}{3 + \sqrt{x^{2} + 5}}$ .

$\frac{(2 + x) (2 - x)}{3 - \sqrt{x^{2} + 5}} \cdot \frac{3 + \sqrt{x^{2} + 5}}{3 + \sqrt{x^{2} + 5}}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{(2 + x) (2 - x)}{3 - \sqrt{x^{2} + 5}}$ by $\frac{3 + \sqrt{x^{2} + 5}}{3 + \sqrt{x^{2} + 5}}$ .

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{(3 - \sqrt{x^{2} + 5}) (3 + \sqrt{x^{2} + 5})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{9 + 3 \sqrt{x^{2} + 5} - 3 \sqrt{x^{2} + 5} - {\sqrt{x^{2} + 5}}^{2}}$

Step 3.3

Simplify.

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{- x^{2} + 4}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{- x^{2} + 2^{2}}$

Step 4.2

Reorder $- x^{2}$ and $2^{2}$ .

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{2^{2} - x^{2}}$

Step 4.3

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

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{(2 + x) (2 - x)}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $2 + x$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(2 + x) (2 - x) (3 + \sqrt{x^{2} + 5})}{(2 + x) (2 - x)}$

Step 5.1.2

Rewrite the expression.

$\frac{(2 - x) (3 + \sqrt{x^{2} + 5})}{2 - x}$

Step 5.2

Cancel the common factor of $2 - x$ .

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Step 5.2.1

Cancel the common factor.

$\frac{(2 - x) (3 + \sqrt{x^{2} + 5})}{2 - x}$

Step 5.2.2

Divide $3 + \sqrt{x^{2} + 5}$ by $1$ .

$3 + \sqrt{x^{2} + 5}$