Calculus Examples

$\frac{\frac{25}{a} - a}{5 + a}$

Step 1

Multiply the numerator and denominator of the fraction by $a$ .

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Step 1.1

Multiply $\frac{\frac{25}{a} - a}{5 + a}$ by $\frac{a}{a}$ .

$\frac{a}{a} \cdot \frac{\frac{25}{a} - a}{5 + a}$

Step 1.2

Combine.

$\frac{a (\frac{25}{a} - a)}{a (5 + a)}$

Step 2

Apply the distributive property.

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

Step 3

Cancel the common factor of $a$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Rewrite $a (a)$ as $a^{2}$ .

$\frac{5^{2} - a^{2}}{a \cdot 5 + a \cdot a}$

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

$\frac{(5 + a) (5 - a)}{a \cdot 5 + a \cdot a}$

Step 5

Simplify terms.

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Step 5.1

Factor $a$ out of $a \cdot 5 + a \cdot a$ .

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Step 5.1.1

Factor $a$ out of $a \cdot 5$ .

$\frac{(5 + a) (5 - a)}{a (5) + a \cdot a}$

Step 5.1.2

Factor $a$ out of $a \cdot a$ .

$\frac{(5 + a) (5 - a)}{a (5) + a (a)}$

Step 5.1.3

Factor $a$ out of $a (5) + a (a)$ .

$\frac{(5 + a) (5 - a)}{a (5 + a)}$

Step 5.2

Cancel the common factor of $5 + a$ .

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Step 5.2.1

Cancel the common factor.

$\frac{(5 + a) (5 - a)}{a (5 + a)}$

Step 5.2.2

Rewrite the expression.

$\frac{5 - a}{a}$