Algebra Examples

$\frac{4}{x^{2} - 8 x + 12} = \frac{x}{x - 2} + \frac{1}{x - 6}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{x}{x - 2}$ from both sides of the equation.

$\frac{4}{x^{2} - 8 x + 12} - \frac{x}{x - 2} = \frac{1}{x - 6}$

Step 1.2

Subtract $\frac{1}{x - 6}$ from both sides of the equation.

$\frac{4}{x^{2} - 8 x + 12} - \frac{x}{x - 2} - \frac{1}{x - 6} = 0$

Step 2

Simplify $\frac{4}{x^{2} - 8 x + 12} - \frac{x}{x - 2} - \frac{1}{x - 6}$ .

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Step 2.1

Factor $x^{2} - 8 x + 12$ 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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 2.1.2

Write the factored form using these integers.

$\frac{4}{(x - 6) (x - 2)} - \frac{x}{x - 2} - \frac{1}{x - 6} = 0$

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{x}{x - 2}$ by $\frac{x - 6}{x - 6}$ .

$\frac{4}{(x - 6) (x - 2)} - (\frac{x}{x - 2} \cdot \frac{x - 6}{x - 6}) - \frac{1}{x - 6} = 0$

Step 2.2.2

Multiply $\frac{x}{x - 2}$ by $\frac{x - 6}{x - 6}$ .

$\frac{4}{(x - 6) (x - 2)} - \frac{x (x - 6)}{(x - 2) (x - 6)} - \frac{1}{x - 6} = 0$

Step 2.2.3

Multiply $\frac{1}{x - 6}$ by $\frac{x - 2}{x - 2}$ .

$\frac{4}{(x - 6) (x - 2)} - \frac{x (x - 6)}{(x - 2) (x - 6)} - (\frac{1}{x - 6} \cdot \frac{x - 2}{x - 2}) = 0$

Step 2.2.4

Multiply $\frac{1}{x - 6}$ by $\frac{x - 2}{x - 2}$ .

$\frac{4}{(x - 6) (x - 2)} - \frac{x (x - 6)}{(x - 2) (x - 6)} - \frac{x - 2}{(x - 6) (x - 2)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{4 - x (x - 6) - (x - 2)}{(x - 2) (x - 6)} = 0$

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

$\frac{4 - x \cdot x - x \cdot - 6 - (x - 2)}{(x - 2) (x - 6)} = 0$

Step 2.4.2

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

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Step 2.4.2.1

Move $x$ .

$\frac{4 - (x \cdot x) - x \cdot - 6 - (x - 2)}{(x - 2) (x - 6)} = 0$

Step 2.4.2.2

Multiply $x$ by $x$ .

$\frac{4 - x^{2} - x \cdot - 6 - (x - 2)}{(x - 2) (x - 6)} = 0$

Step 2.4.3

Multiply $- 6$ by $- 1$ .

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

Step 2.4.4

Apply the distributive property.

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

Step 2.4.5

Multiply $- 1$ by $- 2$ .

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

Step 2.5

Add $4$ and $2$ .

$\frac{- x^{2} + 6 x - x + 6}{(x - 2) (x - 6)} = 0$

Step 2.6

Subtract $x$ from $6 x$ .

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

Step 2.7

Factor by grouping.

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Step 2.7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 5$ .

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Step 2.7.1.1

Factor $5$ out of $5 x$ .

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

Step 2.7.1.2

Rewrite $5$ as $- 1$ plus $6$

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

Step 2.7.1.3

Apply the distributive property.

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

Step 2.7.2

Factor out the greatest common factor from each group.

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Step 2.7.2.1

Group the first two terms and the last two terms.

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

Step 2.7.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 1) - 6 (- x - 1)}{(x - 2) (x - 6)} = 0$

Step 2.7.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{(- x - 1) (x - 6)}{(x - 2) (x - 6)} = 0$

Step 2.8

Cancel the common factor of $x - 6$ .

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Step 2.8.1

Cancel the common factor.

$\frac{(- x - 1) (x - 6)}{(x - 2) (x - 6)} = 0$

Step 2.8.2

Rewrite the expression.

$\frac{- x - 1}{x - 2} = 0$

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Factor $- 1$ out of $- (x) - 1 (1)$ .

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

Step 2.12

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

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

Step 2.13

Move the negative in front of the fraction.

$- \frac{x + 1}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$