Calculus Examples

$\frac{24}{(x + 4) (x - 4)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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 $A$ .

$\frac{A}{x + 4}$

Step 1.2

$\frac{A}{x + 4} + \frac{B}{x - 4}$

Step 1.3

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

$\frac{24 (x + 4) (x - 4)}{(x + 4) (x - 4)} = \frac{(A) (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.4

Cancel the common factor of $x + 4$ .

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Step 1.4.1

Cancel the common factor.

$\frac{24 (x + 4) (x - 4)}{(x + 4) (x - 4)} = \frac{(A) (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.4.2

Rewrite the expression.

$\frac{24 (x - 4)}{x - 4} = \frac{(A) (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.5

Cancel the common factor of $x - 4$ .

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Step 1.5.1

Cancel the common factor.

$\frac{24 (x - 4)}{x - 4} = \frac{(A) (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.5.2

Divide $24$ by $1$ .

$24 = \frac{(A) (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x + 4$ .

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Step 1.6.1.1

Cancel the common factor.

$24 = \frac{A (x + 4) (x - 4)}{x + 4} + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6.1.2

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

$24 = (A) (x - 4) + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6.2

Apply the distributive property.

$24 = A x + A \cdot - 4 + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6.3

Move $- 4$ to the left of $A$ .

$24 = A x - 4 \cdot A + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6.4

Cancel the common factor of $x - 4$ .

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Step 1.6.4.1

Cancel the common factor.

$24 = A x - 4 A + \frac{(B) (x + 4) (x - 4)}{x - 4}$

Step 1.6.4.2

Divide $(B) (x + 4)$ by $1$ .

$24 = A x - 4 A + (B) (x + 4)$

Step 1.6.5

Apply the distributive property.

$24 = A x - 4 A + B x + B \cdot 4$

Step 1.6.6

Move $4$ to the left of $B$ .

$24 = A x - 4 A + B x + 4 B$

Step 1.7

Move $- 4 A$ .

$24 = A x + B x - 4 A + 4 B$

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.

$0 = A + B$

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.

$24 = - 4 A + 4 B$

Step 2.3

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

$0 = A + B$

$24 = - 4 A + 4 B$

$0 = A + B$

$24 = - 4 A + 4 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $0 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 0$ .

$A + B = 0$

$24 = - 4 A + 4 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$24 = - 4 A + 4 B$

$A = - B$

$24 = - 4 A + 4 B$

Step 3.2

Replace all occurrences of $A$ with $- B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $24 = - 4 A + 4 B$ with $- B$ .

$24 = - 4 (- B) + 4 B$

$A = - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 4 (- B) + 4 B$ .

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Step 3.2.2.1.1

Multiply $- 1$ by $- 4$ .

$24 = 4 B + 4 B$

$A = - B$

Step 3.2.2.1.2

Add $4 B$ and $4 B$ .

$24 = 8 B$

$A = - B$

$24 = 8 B$

$A = - B$

$24 = 8 B$

$A = - B$

$24 = 8 B$

$A = - B$

Step 3.3

Solve for $B$ in $24 = 8 B$ .

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Step 3.3.1

Rewrite the equation as $8 B = 24$ .

$8 B = 24$

$A = - B$

Step 3.3.2

Divide each term in $8 B = 24$ by $8$ and simplify.

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Step 3.3.2.1

Divide each term in $8 B = 24$ by $8$ .

$\frac{8 B}{8} = \frac{24}{8}$

$A = - B$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{8 B}{8} = \frac{24}{8}$

$A = - B$

Step 3.3.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{24}{8}$

$A = - B$

$B = \frac{24}{8}$

$A = - B$

$B = \frac{24}{8}$

$A = - B$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $24$ by $8$ .

$B = 3$

$A = - B$

$B = 3$

$A = - B$

$B = 3$

$A = - B$

$B = 3$

$A = - B$

Step 3.4

Replace all occurrences of $B$ with $3$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = - B$ with $3$ .

$A = - (3)$

$B = 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 1$ by $3$ .

$A = - 3$

$B = 3$

$A = - 3$

$B = 3$

$A = - 3$

$B = 3$

Step 3.5

List all of the solutions.

$A = - 3, B = 3$

Step 4

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

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

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