Algebra Examples

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

Step 1

Subtract

$\frac{2}{x^{2} - 6 x + 8}$ from both sides of the equation.

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

Step 2

Simplify

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

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Step 2.1

Factor

$x^{2} - 6 x + 8$ 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

$8$ and whose sum is

$- 6$ .

$- 4, - 2$

Step 2.1.2

Write the factored form using these integers.

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

Step 2.2

To write

$\frac{x}{x - 2}$ as a fraction with a common denominator, multiply by

$\frac{x - 4}{x - 4}$ .

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

Step 2.3

To write

$\frac{1}{x - 4}$ as a fraction with a common denominator, multiply by

$\frac{x - 2}{x - 2}$ .

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

Step 2.4

Write each expression with a common denominator of

$(x - 2) (x - 4)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 2.4.1

Multiply

$\frac{x}{x - 2}$ by

$\frac{x - 4}{x - 4}$ .

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

Step 2.4.2

Multiply

$\frac{1}{x - 4}$ by

$\frac{x - 2}{x - 2}$ .

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

Step 2.4.3

Reorder the factors of

$(x - 4) (x - 2)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Multiply

$x$ by

$x$ .

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

Step 2.7.3

Move

$- 4$ to the left of

$x$ .

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

Step 2.8

Add

$- 4 x$ and

$x$ .

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

Step 2.9

Subtract

$2$ from

$- 2$ .

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

Step 2.10

Factor

$x^{2} - 3 x - 4$ using the AC method.

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Step 2.10.1

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

$- 3$ .

$- 4, 1$

Step 2.10.2

Write the factored form using these integers.

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

Step 2.11

Cancel the common factor of

$x - 4$ .

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Step 2.11.1

Cancel the common factor.

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

Step 2.11.2

Rewrite the expression.

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

Step 3

Set the numerator equal to zero.

$x + 1 = 0$

Step 4

Subtract

$1$ from both sides of the equation.

$x = - 1$