Basic Math Examples

$\frac{a}{a^{2} - 36} + \frac{a - 10}{a^{2} - 2 a - 24}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{a}{a^{2} - 6^{2}} + \frac{a - 10}{a^{2} - 2 a - 24}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 6$ .

$\frac{a}{(a + 6) (a - 6)} + \frac{a - 10}{a^{2} - 2 a - 24}$

Step 1.2

Factor $a^{2} - 2 a - 24$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{a}{(a + 6) (a - 6)} + \frac{a - 10}{(a - 6) (a + 4)}$

Step 2

To write $\frac{a}{(a + 6) (a - 6)}$ as a fraction with a common denominator, multiply by $\frac{a + 4}{a + 4}$ .

$\frac{a}{(a + 6) (a - 6)} \cdot \frac{a + 4}{a + 4} + \frac{a - 10}{(a - 6) (a + 4)}$

Step 3

To write $\frac{a - 10}{(a - 6) (a + 4)}$ as a fraction with a common denominator, multiply by $\frac{a + 6}{a + 6}$ .

$\frac{a}{(a + 6) (a - 6)} \cdot \frac{a + 4}{a + 4} + \frac{a - 10}{(a - 6) (a + 4)} \cdot \frac{a + 6}{a + 6}$

Step 4

Write each expression with a common denominator of $(a + 6) (a - 6) (a + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{(a + 6) (a - 6)}$ by $\frac{a + 4}{a + 4}$ .

$\frac{a (a + 4)}{(a + 6) (a - 6) (a + 4)} + \frac{a - 10}{(a - 6) (a + 4)} \cdot \frac{a + 6}{a + 6}$

Step 4.2

Multiply $\frac{a - 10}{(a - 6) (a + 4)}$ by $\frac{a + 6}{a + 6}$ .

$\frac{a (a + 4)}{(a + 6) (a - 6) (a + 4)} + \frac{(a - 10) (a + 6)}{(a - 6) (a + 4) (a + 6)}$

Step 4.3

Reorder the factors of $(a + 6) (a - 6) (a + 4)$ .

$\frac{a (a + 4)}{(a + 4) (a + 6) (a - 6)} + \frac{(a - 10) (a + 6)}{(a - 6) (a + 4) (a + 6)}$

Step 4.4

Reorder the factors of $(a - 6) (a + 4) (a + 6)$ .

$\frac{a (a + 4)}{(a + 4) (a + 6) (a - 6)} + \frac{(a - 10) (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a (a + 4) + (a - 10) (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot a + a \cdot 4 + (a - 10) (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot 4 + (a - 10) (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6.3

Move $4$ to the left of $a$ .

$\frac{a^{2} + 4 \cdot a + (a - 10) (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6.4

Expand $(a - 10) (a + 6)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

$\frac{a^{2} + 4 a + a (a + 6) - 10 (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6.4.2

Apply the distributive property.

$\frac{a^{2} + 4 a + a \cdot a + a \cdot 6 - 10 (a + 6)}{(a + 4) (a + 6) (a - 6)}$

Step 6.4.3

Apply the distributive property.

$\frac{a^{2} + 4 a + a \cdot a + a \cdot 6 - 10 a - 10 \cdot 6}{(a + 4) (a + 6) (a - 6)}$

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

Multiply $a$ by $a$ .

$\frac{a^{2} + 4 a + a^{2} + a \cdot 6 - 10 a - 10 \cdot 6}{(a + 4) (a + 6) (a - 6)}$

Step 6.5.1.2

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

$\frac{a^{2} + 4 a + a^{2} + 6 \cdot a - 10 a - 10 \cdot 6}{(a + 4) (a + 6) (a - 6)}$

Step 6.5.1.3

Multiply $- 10$ by $6$ .

$\frac{a^{2} + 4 a + a^{2} + 6 a - 10 a - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.5.2

Subtract $10 a$ from $6 a$ .

$\frac{a^{2} + 4 a + a^{2} - 4 a - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.6

Add $a^{2}$ and $a^{2}$ .

$\frac{2 a^{2} + 4 a - 4 a - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.7

Subtract $4 a$ from $4 a$ .

$\frac{2 a^{2} + 0 - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.8

Add $2 a^{2}$ and $0$ .

$\frac{2 a^{2} - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.9

Factor $2$ out of $2 a^{2} - 60$ .

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Step 6.9.1

Factor $2$ out of $2 a^{2}$ .

$\frac{2 (a^{2}) - 60}{(a + 4) (a + 6) (a - 6)}$

Step 6.9.2

Factor $2$ out of $- 60$ .

$\frac{2 a^{2} + 2 \cdot - 30}{(a + 4) (a + 6) (a - 6)}$

Step 6.9.3

Factor $2$ out of $2 a^{2} + 2 \cdot - 30$ .

$\frac{2 (a^{2} - 30)}{(a + 4) (a + 6) (a - 6)}$