Precalculus Examples

$\sqrt{25 - x^{2}} + \frac{x^{2}}{\sqrt{25 - x^{2}}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} - x^{2}} + \frac{x^{2}}{\sqrt{25 - x^{2}}}$

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 $a = 5$ and $b = x$ .

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $25$ as $5^{2}$ .

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

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

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

Step 1.4

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

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

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

Step 1.5.2

Raise $\sqrt{(5 + x) (5 - x)}$ to the power of $1$ .

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1} \sqrt{(5 + x) (5 - x)}}$

Step 1.5.3

Raise $\sqrt{(5 + x) (5 - x)}$ to the power of $1$ .

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1} {\sqrt{(5 + x) (5 - x)}}^{1}}$

Step 1.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1 + 1}}$

Step 1.5.5

Add $1$ and $1$ .

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

Step 1.5.6

Rewrite ${\sqrt{(5 + x) (5 - x)}}^{2}$ as $(5 + x) (5 - x)$ .

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Step 1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + x) (5 - x)}$ as ${((5 + x) (5 - x))}^{\frac{1}{2}}$ .

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{({((5 + x) (5 - x))}^{\frac{1}{2}})}^{2}}$

Step 1.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{\frac{1}{2} \cdot 2}}$

Step 1.5.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

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

Step 1.5.6.4.2

Rewrite the expression.

$\sqrt{(5 + x) (5 - x)} + \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{1}}$

Step 1.5.6.5

Simplify.

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

Step 2

To write $\sqrt{(5 + x) (5 - x)}$ as a fraction with a common denominator, multiply by $\frac{(5 + x) (5 - x)}{(5 + x) (5 - x)}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine $\sqrt{(5 + x) (5 - x)}$ and $\frac{(5 + x) (5 - x)}{(5 + x) (5 - x)}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $\sqrt{(5 + x) (5 - x)}$ out of $\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) + x^{2} \sqrt{(5 + x) (5 - x)}$ .

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Step 4.1.1

Factor $\sqrt{(5 + x) (5 - x)}$ out of $x^{2} \sqrt{(5 + x) (5 - x)}$ .

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

Step 4.1.2

Factor $\sqrt{(5 + x) (5 - x)}$ out of $\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) + \sqrt{(5 + x) (5 - x)} x^{2}$ .

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

Step 4.2

Expand $(5 + x) (5 - x)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $5$ by $5$ .

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

Step 4.3.1.2

Multiply $- 1$ by $5$ .

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.5.1

Move $x$ .

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

Step 4.3.1.5.2

Multiply $x$ by $x$ .

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

Step 4.3.2

Add $- 5 x$ and $5 x$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 + 0 - x^{2} + x^{2})}{(5 + x) (5 - x)}$

Step 4.3.3

Add $25$ and $0$ .

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

Step 4.4

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

$\frac{\sqrt{(5 + x) (5 - x)} (25 + 0)}{(5 + x) (5 - x)}$

Step 4.5

Add $25$ and $0$ .

$\frac{\sqrt{(5 + x) (5 - x)} \cdot 25}{(5 + x) (5 - x)}$

Step 5

Move $25$ to the left of $\sqrt{(5 + x) (5 - x)}$ .

$\frac{25 \sqrt{(5 + x) (5 - x)}}{(5 + x) (5 - x)}$