Basic Math Examples

$\frac{a}{a^{2} - 16} - \frac{3}{a^{2} + 5 a + 4}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 1.2

Factor $a^{2} + 5 a + 4$ 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 $4$ and whose sum is $5$ .

$1, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{a}{(a + 4) (a - 4)} - \frac{3}{(a + 1) (a + 4)}$

Step 2

To write $\frac{a}{(a + 4) (a - 4)}$ as a fraction with a common denominator, multiply by $\frac{a + 1}{a + 1}$ .

$\frac{a}{(a + 4) (a - 4)} \cdot \frac{a + 1}{a + 1} - \frac{3}{(a + 1) (a + 4)}$

Step 3

To write $- \frac{3}{(a + 1) (a + 4)}$ as a fraction with a common denominator, multiply by $\frac{a - 4}{a - 4}$ .

$\frac{a}{(a + 4) (a - 4)} \cdot \frac{a + 1}{a + 1} - \frac{3}{(a + 1) (a + 4)} \cdot \frac{a - 4}{a - 4}$

Step 4

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

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Step 4.1

Multiply $\frac{a}{(a + 4) (a - 4)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{a (a + 1)}{(a + 4) (a - 4) (a + 1)} - \frac{3}{(a + 1) (a + 4)} \cdot \frac{a - 4}{a - 4}$

Step 4.2

Multiply $\frac{3}{(a + 1) (a + 4)}$ by $\frac{a - 4}{a - 4}$ .

$\frac{a (a + 1)}{(a + 4) (a - 4) (a + 1)} - \frac{3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 4.3

Reorder the factors of $(a + 4) (a - 4) (a + 1)$ .

$\frac{a (a + 1)}{(a + 1) (a + 4) (a - 4)} - \frac{3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a (a + 1) - 3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot a + a \cdot 1 - 3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 6.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot 1 - 3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 6.3

Multiply $a$ by $1$ .

$\frac{a^{2} + a - 3 (a - 4)}{(a + 1) (a + 4) (a - 4)}$

Step 6.4

Apply the distributive property.

$\frac{a^{2} + a - 3 a - 3 \cdot - 4}{(a + 1) (a + 4) (a - 4)}$

Step 6.5

Multiply $- 3$ by $- 4$ .

$\frac{a^{2} + a - 3 a + 12}{(a + 1) (a + 4) (a - 4)}$

Step 6.6

Subtract $3 a$ from $a$ .

$\frac{a^{2} - 2 a + 12}{(a + 1) (a + 4) (a - 4)}$