Algebra Examples

$\frac{a}{a - 6} - \frac{3}{a + 6} + \frac{a^{2}}{36 - a^{2}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

$\frac{a}{a - 6} - \frac{3}{a + 6} + \frac{a^{2}}{6^{2} - a^{2}}$

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 $a = 6$ and $b = a$ .

$\frac{a}{a - 6} - \frac{3}{a + 6} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 2

To write $\frac{a}{a - 6}$ as a fraction with a common denominator, multiply by $\frac{a + 6}{a + 6}$ .

$\frac{a}{a - 6} \cdot \frac{a + 6}{a + 6} - \frac{3}{a + 6} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 3

To write $- \frac{3}{a + 6}$ as a fraction with a common denominator, multiply by $\frac{a - 6}{a - 6}$ .

$\frac{a}{a - 6} \cdot \frac{a + 6}{a + 6} - \frac{3}{a + 6} \cdot \frac{a - 6}{a - 6} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 4

Write each expression with a common denominator of $(a - 6) (a + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{a - 6}$ by $\frac{a + 6}{a + 6}$ .

$\frac{a (a + 6)}{(a - 6) (a + 6)} - \frac{3}{a + 6} \cdot \frac{a - 6}{a - 6} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 4.2

Multiply $\frac{3}{a + 6}$ by $\frac{a - 6}{a - 6}$ .

$\frac{a (a + 6)}{(a - 6) (a + 6)} - \frac{3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 4.3

Reorder the factors of $(a - 6) (a + 6)$ .

$\frac{a (a + 6)}{(a + 6) (a - 6)} - \frac{3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a (a + 6) - 3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot a + a \cdot 6 - 3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot 6 - 3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6.3

Move $6$ to the left of $a$ .

$\frac{a^{2} + 6 \cdot a - 3 (a - 6)}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6.4

Apply the distributive property.

$\frac{a^{2} + 6 a - 3 a - 3 \cdot - 6}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6.5

Multiply $- 3$ by $- 6$ .

$\frac{a^{2} + 6 a - 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 6.6

Subtract $3 a$ from $6 a$ .

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(6 + a) (6 - a)}$

Step 7

Simplify terms.

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Step 7.1

Reorder terms.

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(a + 6) (6 - a)}$

Step 7.2

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(a + 6) (- 1 (- 6) - a)}$

Step 7.3

Factor $- 1$ out of $- a$ .

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(a + 6) (- 1 (- 6) - (a))}$

Step 7.4

Factor $- 1$ out of $- 1 (- 6) - (a)$ .

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{a^{2}}{(a + 6) (- 1 (- 6 + a))}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Move a negative from the denominator of $\frac{a^{2}}{(a + 6) (- 1 (- 6 + a))}$ to the numerator.

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{- a^{2}}{(a + 6) (- 6 + a)}$

Step 7.5.2

Reorder terms.

$\frac{a^{2} + 3 a + 18}{(a + 6) (a - 6)} + \frac{- a^{2}}{(a + 6) (a - 6)}$

Step 7.6

Combine the numerators over the common denominator.

$\frac{a^{2} + 3 a + 18 - a^{2}}{(a + 6) (a - 6)}$

Step 8

Simplify the numerator.

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Step 8.1

Subtract $a^{2}$ from $a^{2}$ .

$\frac{3 a + 18 + 0}{(a + 6) (a - 6)}$

Step 8.2

Add $3 a + 18$ and $0$ .

$\frac{3 a + 18}{(a + 6) (a - 6)}$

Step 8.3

Factor $3$ out of $3 a + 18$ .

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Step 8.3.1

Factor $3$ out of $3 a$ .

$\frac{3 (a) + 18}{(a + 6) (a - 6)}$

Step 8.3.2

Factor $3$ out of $18$ .

$\frac{3 a + 3 \cdot 6}{(a + 6) (a - 6)}$

Step 8.3.3

Factor $3$ out of $3 a + 3 \cdot 6$ .

$\frac{3 (a + 6)}{(a + 6) (a - 6)}$

Step 9

Cancel the common factor of $a + 6$ .

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Step 9.1

Cancel the common factor.

$\frac{3 (a + 6)}{(a + 6) (a - 6)}$

Step 9.2

Rewrite the expression.

$\frac{3}{a - 6}$