Precalculus Examples

$\frac{x}{8 x^{2} - 10 x + 3}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 8 \cdot 3 = 24$ and whose sum is $b = - 10$ .

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Step 1.1.1.1

Factor $- 10$ out of $- 10 x$ .

$\frac{x}{8 x^{2} - 10 (x) + 3}$

Step 1.1.1.2

Rewrite $- 10$ as $- 4$ plus $- 6$

$\frac{x}{8 x^{2} + (- 4 - 6) x + 3}$

Step 1.1.1.3

Apply the distributive property.

$\frac{x}{8 x^{2} - 4 x - 6 x + 3}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{x}{(8 x^{2} - 4 x) - 6 x + 3}$

Step 1.1.2.2

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

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

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{2 x - 1}$

Step 1.3

$\frac{A}{2 x - 1} + \frac{B}{4 x - 3}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(2 x - 1) (4 x - 3)$ .

$\frac{x (2 x - 1) (4 x - 3)}{(2 x - 1) (4 x - 3)} = \frac{(A) (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.5

Cancel the common factor of $2 x - 1$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x (2 x - 1) (4 x - 3)}{(2 x - 1) (4 x - 3)} = \frac{(A) (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.5.2

Rewrite the expression.

$\frac{x (4 x - 3)}{4 x - 3} = \frac{(A) (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.6

Cancel the common factor of $4 x - 3$ .

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Step 1.6.1

Cancel the common factor.

$\frac{x (4 x - 3)}{4 x - 3} = \frac{(A) (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.6.2

Divide $x$ by $1$ .

$x = \frac{(A) (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $2 x - 1$ .

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Step 1.7.1.1

Cancel the common factor.

$x = \frac{A (2 x - 1) (4 x - 3)}{2 x - 1} + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.1.2

Divide $(A) (4 x - 3)$ by $1$ .

$x = (A) (4 x - 3) + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.2

Apply the distributive property.

$x = A (4 x) + A \cdot - 3 + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.3

Rewrite using the commutative property of multiplication.

$x = 4 A x + A \cdot - 3 + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.4

Move $- 3$ to the left of $A$ .

$x = 4 A x - 3 \cdot A + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.5

Cancel the common factor of $4 x - 3$ .

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Step 1.7.5.1

Cancel the common factor.

$x = 4 A x - 3 A + \frac{(B) (2 x - 1) (4 x - 3)}{4 x - 3}$

Step 1.7.5.2

Divide $(B) (2 x - 1)$ by $1$ .

$x = 4 A x - 3 A + (B) (2 x - 1)$

Step 1.7.6

Apply the distributive property.

$x = 4 A x - 3 A + B (2 x) + B \cdot - 1$

Step 1.7.7

Rewrite using the commutative property of multiplication.

$x = 4 A x - 3 A + 2 B x + B \cdot - 1$

Step 1.7.8

Move $- 1$ to the left of $B$ .

$x = 4 A x - 3 A + 2 B x - 1 \cdot B$

Step 1.7.9

Rewrite $- 1 B$ as $- B$ .

$x = 4 A x - 3 A + 2 B x - B$

Step 1.8

Reorder.

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Step 1.8.1

Move $A$ .

$x = 4 x A - 3 A + 2 B x - B$

Step 1.8.2

Move $B$ .

$x = 4 x A - 3 A + 2 x B - B$

Step 1.8.3

Move $- 3 A$ .

$x = 4 x A + 2 x B - 3 A - B$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = 4 A + 2 B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 3 A - 1 B$

Step 2.3

Set up the system of equations to find the coefficients of the partial fractions.

$1 = 4 A + 2 B$

$0 = - 3 A - 1 B$

$1 = 4 A + 2 B$

$0 = - 3 A - 1 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $1 = 4 A + 2 B$ .

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Step 3.1.1

Rewrite the equation as $4 A + 2 B = 1$ .

$4 A + 2 B = 1$

$0 = - 3 A - 1 B$

Step 3.1.2

Subtract $2 B$ from both sides of the equation.

$4 A = 1 - 2 B$

$0 = - 3 A - 1 B$

Step 3.1.3

Divide each term in $4 A = 1 - 2 B$ by $4$ and simplify.

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Step 3.1.3.1

Divide each term in $4 A = 1 - 2 B$ by $4$ .

$\frac{4 A}{4} = \frac{1}{4} + \frac{- 2 B}{4}$

$0 = - 3 A - 1 B$

Step 3.1.3.2

Simplify the left side.

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Step 3.1.3.2.1

Cancel the common factor of $4$ .

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Step 3.1.3.2.1.1

Cancel the common factor.

$\frac{4 A}{4} = \frac{1}{4} + \frac{- 2 B}{4}$

$0 = - 3 A - 1 B$

Step 3.1.3.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{4} + \frac{- 2 B}{4}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} + \frac{- 2 B}{4}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} + \frac{- 2 B}{4}$

$0 = - 3 A - 1 B$

Step 3.1.3.3

Simplify the right side.

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Step 3.1.3.3.1

Simplify each term.

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Step 3.1.3.3.1.1

Cancel the common factor of $- 2$ and $4$ .

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Step 3.1.3.3.1.1.1

Factor $2$ out of $- 2 B$ .

$A = \frac{1}{4} + \frac{2 (- B)}{4}$

$0 = - 3 A - 1 B$

Step 3.1.3.3.1.1.2

Cancel the common factors.

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Step 3.1.3.3.1.1.2.1

Factor $2$ out of $4$ .

$A = \frac{1}{4} + \frac{2 (- B)}{2 (2)}$

$0 = - 3 A - 1 B$

Step 3.1.3.3.1.1.2.2

Cancel the common factor.

$A = \frac{1}{4} + \frac{2 (- B)}{2 \cdot 2}$

$0 = - 3 A - 1 B$

Step 3.1.3.3.1.1.2.3

Rewrite the expression.

$A = \frac{1}{4} + \frac{- B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} + \frac{- B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} + \frac{- B}{2}$

$0 = - 3 A - 1 B$

Step 3.1.3.3.1.2

Move the negative in front of the fraction.

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - 3 A - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - 3 A - 1 B$

Step 3.2

Replace all occurrences of $A$ with $\frac{1}{4} - \frac{B}{2}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = - 3 A - 1 B$ with $\frac{1}{4} - \frac{B}{2}$ .

$0 = - 3 (\frac{1}{4} - \frac{B}{2}) - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 3 (\frac{1}{4} - \frac{B}{2}) - 1 B$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$0 = - 3 (\frac{1}{4}) - 3 (- \frac{B}{2}) - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.1.2

Combine $- 3$ and $\frac{1}{4}$ .

$0 = \frac{- 3}{4} - 3 (- \frac{B}{2}) - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.1.3

Multiply $- 3 (- \frac{B}{2})$ .

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Step 3.2.2.1.1.3.1

Multiply $- 1$ by $- 3$ .

$0 = \frac{- 3}{4} + 3 (\frac{B}{2}) - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.1.3.2

Combine $3$ and $\frac{B}{2}$ .

$0 = \frac{- 3}{4} + \frac{3 B}{2} - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = \frac{- 3}{4} + \frac{3 B}{2} - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.1.4

Move the negative in front of the fraction.

$0 = - \frac{3}{4} + \frac{3 B}{2} - 1 B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.1.5

Rewrite $- 1 B$ as $- B$ .

$0 = - \frac{3}{4} + \frac{3 B}{2} - B$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{3 B}{2} - B$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.2

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

$0 = - \frac{3}{4} + \frac{3 B}{2} - B \cdot \frac{2}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

Combine $- B$ and $\frac{2}{2}$ .

$0 = - \frac{3}{4} + \frac{3 B}{2} + \frac{- B \cdot 2}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$0 = - \frac{3}{4} + \frac{3 B - B \cdot 2}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{3 B - B \cdot 2}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4

Simplify each term.

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Step 3.2.2.1.4.1

Simplify the numerator.

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Step 3.2.2.1.4.1.1

Factor $B$ out of $3 B - B \cdot 2$ .

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Step 3.2.2.1.4.1.1.1

Factor $B$ out of $3 B$ .

$0 = - \frac{3}{4} + \frac{B \cdot 3 - B \cdot 2}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4.1.1.2

Factor $B$ out of $- B \cdot 2$ .

$0 = - \frac{3}{4} + \frac{B \cdot 3 + B (- 1 \cdot 2)}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4.1.1.3

Factor $B$ out of $B \cdot 3 + B (- 1 \cdot 2)$ .

$0 = - \frac{3}{4} + \frac{B (3 - 1 \cdot 2)}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B (3 - 1 \cdot 2)}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4.1.2

Multiply $- 1$ by $2$ .

$0 = - \frac{3}{4} + \frac{B (3 - 2)}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4.1.3

Subtract $2$ from $3$ .

$0 = - \frac{3}{4} + \frac{B \cdot 1}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B \cdot 1}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.2.2.1.4.2

Multiply $B$ by $1$ .

$0 = - \frac{3}{4} + \frac{B}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$0 = - \frac{3}{4} + \frac{B}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3

Solve for $B$ in $0 = - \frac{3}{4} + \frac{B}{2}$ .

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Step 3.3.1

Rewrite the equation as $- \frac{3}{4} + \frac{B}{2} = 0$ .

$- \frac{3}{4} + \frac{B}{2} = 0$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.2

Add $\frac{3}{4}$ to both sides of the equation.

$\frac{B}{2} = \frac{3}{4}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.3

Multiply both sides of the equation by $2$ .

$2 (\frac{B}{2}) = 2 (\frac{3}{4})$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.4

Simplify both sides of the equation.

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Step 3.3.4.1

Simplify the left side.

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Step 3.3.4.1.1

Cancel the common factor of $2$ .

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Step 3.3.4.1.1.1

Cancel the common factor.

$2 (\frac{B}{2}) = 2 (\frac{3}{4})$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.4.1.1.2

Rewrite the expression.

$B = 2 (\frac{3}{4})$

$A = \frac{1}{4} - \frac{B}{2}$

$B = 2 (\frac{3}{4})$

$A = \frac{1}{4} - \frac{B}{2}$

$B = 2 (\frac{3}{4})$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.4.2

Simplify the right side.

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Step 3.3.4.2.1

Cancel the common factor of $2$ .

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Step 3.3.4.2.1.1

Factor $2$ out of $4$ .

$B = 2 (\frac{3}{2 (2)})$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.4.2.1.2

Cancel the common factor.

$B = 2 (\frac{3}{2 \cdot 2})$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.3.4.2.1.3

Rewrite the expression.

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{B}{2}$

Step 3.4

Replace all occurrences of $B$ with $\frac{3}{2}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = \frac{1}{4} - \frac{B}{2}$ with $\frac{3}{2}$ .

$A = \frac{1}{4} - \frac{\frac{3}{2}}{2}$

$B = \frac{3}{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{1}{4} - \frac{\frac{3}{2}}{2}$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$A = \frac{1}{4} - (\frac{3}{2} \cdot \frac{1}{2})$

$B = \frac{3}{2}$

Step 3.4.2.1.1.2

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 3.4.2.1.1.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$A = \frac{1}{4} - \frac{3}{2 \cdot 2}$

$B = \frac{3}{2}$

Step 3.4.2.1.1.2.2

Multiply $2$ by $2$ .

$A = \frac{1}{4} - \frac{3}{4}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{3}{4}$

$B = \frac{3}{2}$

$A = \frac{1}{4} - \frac{3}{4}$

$B = \frac{3}{2}$

Step 3.4.2.1.2

Simplify terms.

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Step 3.4.2.1.2.1

Combine the numerators over the common denominator.

$A = \frac{1 - 3}{4}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.2

Subtract $3$ from $1$ .

$A = \frac{- 2}{4}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.3

Cancel the common factor of $- 2$ and $4$ .

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Step 3.4.2.1.2.3.1

Factor $2$ out of $- 2$ .

$A = \frac{2 (- 1)}{4}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.3.2

Cancel the common factors.

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Step 3.4.2.1.2.3.2.1

Factor $2$ out of $4$ .

$A = \frac{2 \cdot - 1}{2 \cdot 2}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.3.2.2

Cancel the common factor.

$A = \frac{2 \cdot - 1}{2 \cdot 2}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.3.2.3

Rewrite the expression.

$A = \frac{- 1}{2}$

$B = \frac{3}{2}$

$A = \frac{- 1}{2}$

$B = \frac{3}{2}$

$A = \frac{- 1}{2}$

$B = \frac{3}{2}$

Step 3.4.2.1.2.4

Move the negative in front of the fraction.

$A = - \frac{1}{2}$

$B = \frac{3}{2}$

$A = - \frac{1}{2}$

$B = \frac{3}{2}$

$A = - \frac{1}{2}$

$B = \frac{3}{2}$

$A = - \frac{1}{2}$

$B = \frac{3}{2}$

$A = - \frac{1}{2}$

$B = \frac{3}{2}$

Step 3.5

List all of the solutions.

$A = - \frac{1}{2}, B = \frac{3}{2}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{2 x - 1} + \frac{B}{4 x - 3}$ with the values found for $A$ and $B$ .

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