Algebra Examples

$\frac{4}{x^{2} + x - 6} - \frac{1}{x^{2} - 4} = \frac{2}{x^{2} + 5 x + 6}$

Step 1

Subtract $\frac{2}{x^{2} + 5 x + 6}$ from both sides of the equation.

$\frac{4}{x^{2} + x - 6} - \frac{1}{x^{2} - 4} - \frac{2}{x^{2} + 5 x + 6} = 0$

Step 2

Simplify each term.

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Step 2.1

Factor $x^{2} + x - 6$ using the AC method.

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Step 2.1.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 2.1.2

Write the factored form using these integers.

$\frac{4}{(x - 2) (x + 3)} - \frac{1}{x^{2} - 4} - \frac{2}{x^{2} + 5 x + 6} = 0$

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

$\frac{4}{(x - 2) (x + 3)} - \frac{1}{x^{2} - 2^{2}} - \frac{2}{x^{2} + 5 x + 6} = 0$

Step 2.2.2

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

$\frac{4}{(x - 2) (x + 3)} - \frac{1}{(x + 2) (x - 2)} - \frac{2}{x^{2} + 5 x + 6} = 0$

Step 2.3

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 2.3.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 2.3.2

Write the factored form using these integers.

$\frac{4}{(x - 2) (x + 3)} - \frac{1}{(x + 2) (x - 2)} - \frac{2}{(x + 2) (x + 3)} = 0$

Step 3

Find the common denominator.

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Step 3.1

Multiply $\frac{4}{(x - 2) (x + 3)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{4}{(x - 2) (x + 3)} \cdot \frac{x + 2}{x + 2} - \frac{1}{(x + 2) (x - 2)} - \frac{2}{(x + 2) (x + 3)} = 0$

Step 3.2

Multiply $\frac{4}{(x - 2) (x + 3)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - \frac{1}{(x + 2) (x - 2)} - \frac{2}{(x + 2) (x + 3)} = 0$

Step 3.3

Multiply $\frac{1}{(x + 2) (x - 2)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - (\frac{1}{(x + 2) (x - 2)} \cdot \frac{x + 3}{x + 3}) - \frac{2}{(x + 2) (x + 3)} = 0$

Step 3.4

Multiply $\frac{1}{(x + 2) (x - 2)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - \frac{x + 3}{(x + 2) (x - 2) (x + 3)} - \frac{2}{(x + 2) (x + 3)} = 0$

Step 3.5

Multiply $\frac{2}{(x + 2) (x + 3)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - \frac{x + 3}{(x + 2) (x - 2) (x + 3)} - (\frac{2}{(x + 2) (x + 3)} \cdot \frac{x - 2}{x - 2}) = 0$

Step 3.6

Multiply $\frac{2}{(x + 2) (x + 3)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - \frac{x + 3}{(x + 2) (x - 2) (x + 3)} - \frac{2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 3.7

Reorder the factors of $(x + 2) (x - 2) (x + 3)$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3) (x + 2)} - \frac{x + 3}{(x + 2) (x + 3) (x - 2)} - \frac{2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{4 (x + 2) - (x + 3) - 2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\frac{4 x + 4 \cdot 2 - (x + 3) - 2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5.2

Multiply $4$ by $2$ .

$\frac{4 x + 8 - (x + 3) - 2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5.3

Apply the distributive property.

$\frac{4 x + 8 - x - 1 \cdot 3 - 2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5.4

Multiply $- 1$ by $3$ .

$\frac{4 x + 8 - x - 3 - 2 (x - 2)}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5.5

Apply the distributive property.

$\frac{4 x + 8 - x - 3 - 2 x - 2 \cdot - 2}{(x + 2) (x + 3) (x - 2)} = 0$

Step 5.6

Multiply $- 2$ by $- 2$ .

$\frac{4 x + 8 - x - 3 - 2 x + 4}{(x + 2) (x + 3) (x - 2)} = 0$

Step 6

Subtract $x$ from $4 x$ .

$\frac{3 x + 8 - 3 - 2 x + 4}{(x + 2) (x + 3) (x - 2)} = 0$

Step 7

Subtract $2 x$ from $3 x$ .

$\frac{x + 8 - 3 + 4}{(x + 2) (x + 3) (x - 2)} = 0$

Step 8

Subtract $3$ from $8$ .

$\frac{x + 5 + 4}{(x + 2) (x + 3) (x - 2)} = 0$

Step 9

Add $5$ and $4$ .

$\frac{x + 9}{(x + 2) (x + 3) (x - 2)} = 0$

Step 10

Set the numerator equal to zero.

$x + 9 = 0$

Step 11

Subtract $9$ from both sides of the equation.

$x = - 9$