Algebra Examples

$\frac{1}{(x + 1) (x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2}$

Step 1

Rewrite the equation as $\frac{A}{x + 1} + \frac{B}{x + 2} = \frac{1}{(x + 1) (x + 2)}$ .

$\frac{A}{x + 1} + \frac{B}{x + 2} = \frac{1}{(x + 1) (x + 2)}$

Step 2

Move all terms containing variables to the left side of the equation.

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Step 2.1

Subtract $\frac{1}{(x + 1) (x + 2)}$ from both sides of the equation.

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

Step 2.2

To write $\frac{A}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 2.3

To write $\frac{B}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 2.4

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

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Step 2.4.1

Multiply $\frac{A}{x + 1}$ by $\frac{x + 2}{x + 2}$ .

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

Step 2.4.2

Multiply $\frac{B}{x + 2}$ by $\frac{x + 1}{x + 1}$ .

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

Step 2.4.3

Reorder the factors of $(x + 2) (x + 1)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Move $2$ to the left of $A$ .

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Multiply $B$ by $1$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 3

Set the numerator equal to zero.

$A x + 2 A + B x + B - 1 = 0$

Step 4

Solve the equation for $A$ .

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Step 4.1

Move all terms not containing $A$ to the right side of the equation.

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Step 4.1.1

Subtract $B x$ from both sides of the equation.

$A x + 2 A + B - 1 = - B x$

Step 4.1.2

Subtract $B$ from both sides of the equation.

$A x + 2 A - 1 = - B x - B$

Step 4.1.3

Add $1$ to both sides of the equation.

$A x + 2 A = - B x - B + 1$

Step 4.2

Factor $A$ out of $A x + 2 A$ .

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Step 4.2.1

Factor $A$ out of $A x$ .

$A (x) + 2 A = - B x - B + 1$

Step 4.2.2

Factor $A$ out of $2 A$ .

$A (x) + A \cdot 2 = - B x - B + 1$

Step 4.2.3

Factor $A$ out of $A (x) + A \cdot 2$ .

$A (x + 2) = - B x - B + 1$

Step 4.3

Divide each term in $A (x + 2) = - B x - B + 1$ by $x + 2$ and simplify.

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Step 4.3.1

Divide each term in $A (x + 2) = - B x - B + 1$ by $x + 2$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x + 2$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{- B x}{x + 2} + \frac{- B}{x + 2} + \frac{1}{x + 2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Move the negative in front of the fraction.

$A = - \frac{B x}{x + 2} + \frac{- B}{x + 2} + \frac{1}{x + 2}$

Step 4.3.3.1.2

Move the negative in front of the fraction.

$A = - \frac{B x}{x + 2} - \frac{B}{x + 2} + \frac{1}{x + 2}$

Step 4.3.3.2

Simplify terms.

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Step 4.3.3.2.1

Combine the numerators over the common denominator.

$A = \frac{- B x - B}{x + 2} + \frac{1}{x + 2}$

Step 4.3.3.2.2

Combine the numerators over the common denominator.

$A = \frac{- B x - B + 1}{x + 2}$

Step 4.3.3.2.3

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

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

Step 4.3.3.2.4

Factor $- 1$ out of $- B$ .

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

Step 4.3.3.2.5

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

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

Step 4.3.3.2.6

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

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

Step 4.3.3.2.7

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

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

Step 4.3.3.2.8

Simplify the expression.

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Step 4.3.3.2.8.1

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

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

Step 4.3.3.2.8.2

Move the negative in front of the fraction.

$A = - \frac{B x + B - 1}{x + 2}$