Algebra Examples

$\frac{7 x + 5}{x^{2} + 8 x + 12} - \frac{7}{x + 2}$

Step 1

Factor $x^{2} + 8 x + 12$ using the AC method.

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Step 1.1

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 $12$ and whose sum is $8$ .

$2, 6$

Step 1.2

Write the factored form using these integers.

$\frac{7 x + 5}{(x + 2) (x + 6)} - \frac{7}{x + 2}$

Step 2

To write $- \frac{7}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{7 x + 5}{(x + 2) (x + 6)} - \frac{7}{x + 2} \cdot \frac{x + 6}{x + 6}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{7}{x + 2}$ by $\frac{x + 6}{x + 6}$ .

$\frac{7 x + 5}{(x + 2) (x + 6)} - \frac{7 (x + 6)}{(x + 2) (x + 6)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{7 x + 5 - 7 (x + 6)}{(x + 2) (x + 6)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{7 x + 5 - 7 x - 7 \cdot 6}{(x + 2) (x + 6)}$

Step 4.2

Multiply $- 7$ by $6$ .

$\frac{7 x + 5 - 7 x - 42}{(x + 2) (x + 6)}$

Step 4.3

Subtract $7 x$ from $7 x$ .

$\frac{0 + 5 - 42}{(x + 2) (x + 6)}$

Step 4.4

Add $0$ and $5$ .

$\frac{5 - 42}{(x + 2) (x + 6)}$

Step 4.5

Subtract $42$ from $5$ .

$\frac{- 37}{(x + 2) (x + 6)}$

Step 5

Move the negative in front of the fraction.

$- \frac{37}{(x + 2) (x + 6)}$