Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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 $- 1$ .

$- 3, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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 $- 5$ .

$- 3, - 2$

Step 1.3.2

Write the factored form using these integers.

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

Step 2

To write $- \frac{2 x}{(x - 3) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

To write $\frac{4}{(x - 3) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 4

Write each expression with a common denominator of $(x - 3) (x + 2) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $- 2 x (x - 2) + 4 (x + 2)$ .

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Step 6.1.1

Factor $2$ out of $- 2 x (x - 2)$ .

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

Step 6.1.2

Factor $2$ out of $4 (x + 2)$ .

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

Step 6.1.3

Factor $2$ out of $2 (- x (x - 2)) + 2 (2 (x + 2))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1

Move $x$ .

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

Step 6.3.2

Multiply $x$ by $x$ .

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

Step 6.4

Multiply $- 2$ by $- 1$ .

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Multiply $2$ by $2$ .

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

Step 6.7

Add $2 x$ and $2 x$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- x^{2}$ .

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

Step 7.2

Factor $- 1$ out of $4 x$ .

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

Step 7.3

Factor $- 1$ out of $- (x^{2}) - (- 4 x)$ .

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

Step 7.4

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 7.5

Factor $- 1$ out of $- (x^{2} - 4 x) - 1 (- 4)$ .

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

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (x^{2} - 4 x - 4)$ as $- 1 (x^{2} - 4 x - 4)$ .

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

Step 7.6.2

Move the negative in front of the fraction.

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