Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Raise $x - 2$ to the power of $1$ .

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

Step 2.3.3

Raise $x - 2$ to the power of $1$ .

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

Step 2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

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

Step 2.7.2

Multiply $2$ by $- 1$ .

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

Step 2.7.3

Multiply $- 1$ by $- 1$ .

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

Step 3

Set the numerator equal to zero.

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

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 $B$ from both sides of the equation.

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

Step 4.1.2

Add $2 x$ to both sides of the equation.

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

Step 4.1.3

Subtract $1$ from both sides of the equation.

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

Step 4.2

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

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Step 4.2.1

Factor $A$ out of $A x$ .

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

Step 4.2.2

Factor $A$ out of $- 2 A$ .

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

Step 4.2.3

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

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

Step 4.3

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

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Step 4.3.1

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

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

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.

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

Step 4.3.2.1.2

Divide $A$ by $1$ .

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

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.

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

Step 4.3.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.3.2

Simplify terms.

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Step 4.3.3.2.1

Combine the numerators over the common denominator.

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

Step 4.3.3.2.2

Combine the numerators over the common denominator.

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

Step 4.3.3.2.3

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

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

Step 4.3.3.2.4

Factor $- 1$ out of $2 x$ .

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

Step 4.3.3.2.5

Factor $- 1$ out of $- (B) - (- 2 x)$ .

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

Step 4.3.3.2.6

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

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

Step 4.3.3.2.7

Factor $- 1$ out of $- (B - 2 x) - 1 (1)$ .

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

Step 4.3.3.2.8

Simplify the expression.

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Step 4.3.3.2.8.1

Rewrite $- (B - 2 x + 1)$ as $- 1 (B - 2 x + 1)$ .

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

Step 4.3.3.2.8.2

Move the negative in front of the fraction.

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