Precalculus Examples

$\frac{3 x + 50}{(x - 9) (x + 2)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

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 $A$ .

$\frac{A}{x - 9}$

Step 1.2

$\frac{A}{x - 9} + \frac{B}{x + 2}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 9) (x + 2)$ .

$\frac{(3 x + 50) (x - 9) (x + 2)}{(x - 9) (x + 2)} = \frac{(A) (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.4

Cancel the common factor of $x - 9$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$\frac{(3 x + 50) (x - 9) (x + 2)}{(x - 9) (x + 2)} = \frac{(A) (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.4.2

Rewrite the expression.

$\frac{(3 x + 50) (x + 2)}{x + 2} = \frac{(A) (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.5

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

$\frac{(3 x + 50) (x + 2)}{x + 2} = \frac{(A) (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.5.2

Divide $3 x + 50$ by $1$ .

$3 x + 50 = \frac{(A) (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6

Simplify each term.

Tap for more steps...

Step 1.6.1

Cancel the common factor of $x - 9$ .

Tap for more steps...

Step 1.6.1.1

Cancel the common factor.

$3 x + 50 = \frac{A (x - 9) (x + 2)}{x - 9} + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6.1.2

Divide $(A) (x + 2)$ by $1$ .

$3 x + 50 = (A) (x + 2) + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6.2

Apply the distributive property.

$3 x + 50 = A x + A \cdot 2 + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6.3

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

$3 x + 50 = A x + 2 \cdot A + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6.4

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 1.6.4.1

Cancel the common factor.

$3 x + 50 = A x + 2 A + \frac{(B) (x - 9) (x + 2)}{x + 2}$

Step 1.6.4.2

Divide $(B) (x - 9)$ by $1$ .

$3 x + 50 = A x + 2 A + (B) (x - 9)$

Step 1.6.5

Apply the distributive property.

$3 x + 50 = A x + 2 A + B x + B \cdot - 9$

Step 1.6.6

Move $- 9$ to the left of $B$ .

$3 x + 50 = A x + 2 A + B x - 9 B$

Step 1.7

Move $2 A$ .

$3 x + 50 = A x + B x + 2 A - 9 B$

Step 2

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

Tap for more steps...

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.

$3 = A + B$

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.

$50 = 2 A - 9 B$

Step 2.3

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

$3 = A + B$

$50 = 2 A - 9 B$

$3 = A + B$

$50 = 2 A - 9 B$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Solve for $A$ in $3 = A + B$ .

Tap for more steps...

Step 3.1.1

Rewrite the equation as $A + B = 3$ .

$A + B = 3$

$50 = 2 A - 9 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 3 - B$

$50 = 2 A - 9 B$

$A = 3 - B$

$50 = 2 A - 9 B$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Replace all occurrences of $A$ in $50 = 2 A - 9 B$ with $3 - B$ .

$50 = 2 (3 - B) - 9 B$

$A = 3 - B$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (3 - B) - 9 B$ .

Tap for more steps...

Step 3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1.1

Apply the distributive property.

$50 = 2 \cdot 3 + 2 (- B) - 9 B$

$A = 3 - B$

Step 3.2.2.1.1.2

Multiply $2$ by $3$ .

$50 = 6 + 2 (- B) - 9 B$

$A = 3 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $2$ .

$50 = 6 - 2 B - 9 B$

$A = 3 - B$

$50 = 6 - 2 B - 9 B$

$A = 3 - B$

Step 3.2.2.1.2

Subtract $9 B$ from $- 2 B$ .

$50 = 6 - 11 B$

$A = 3 - B$

$50 = 6 - 11 B$

$A = 3 - B$

$50 = 6 - 11 B$

$A = 3 - B$

$50 = 6 - 11 B$

$A = 3 - B$

Step 3.3

Solve for $B$ in $50 = 6 - 11 B$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $6 - 11 B = 50$ .

$6 - 11 B = 50$

$A = 3 - B$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Subtract $6$ from both sides of the equation.

$- 11 B = 50 - 6$

$A = 3 - B$

Step 3.3.2.2

Subtract $6$ from $50$ .

$- 11 B = 44$

$A = 3 - B$

$- 11 B = 44$

$A = 3 - B$

Step 3.3.3

Divide each term in $- 11 B = 44$ by $- 11$ and simplify.

Tap for more steps...

Step 3.3.3.1

Divide each term in $- 11 B = 44$ by $- 11$ .

$\frac{- 11 B}{- 11} = \frac{44}{- 11}$

$A = 3 - B$

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $- 11$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 11 B}{- 11} = \frac{44}{- 11}$

$A = 3 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{44}{- 11}$

$A = 3 - B$

$B = \frac{44}{- 11}$

$A = 3 - B$

$B = \frac{44}{- 11}$

$A = 3 - B$

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Divide $44$ by $- 11$ .

$B = - 4$

$A = 3 - B$

$B = - 4$

$A = 3 - B$

$B = - 4$

$A = 3 - B$

$B = - 4$

$A = 3 - B$

Step 3.4

Replace all occurrences of $B$ with $- 4$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $B$ in $A = 3 - B$ with $- 4$ .

$A = 3 - (- 4)$

$B = - 4$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $3 - (- 4)$ .

Tap for more steps...

Step 3.4.2.1.1

Multiply $- 1$ by $- 4$ .

$A = 3 + 4$

$B = - 4$

Step 3.4.2.1.2

Add $3$ and $4$ .

$A = 7$

$B = - 4$

$A = 7$

$B = - 4$

$A = 7$

$B = - 4$

$A = 7$

$B = - 4$

Step 3.5

List all of the solutions.

$A = 7, B = - 4$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 9} + \frac{B}{x + 2}$ with the values found for $A$ and $B$ .

$\frac{7}{x - 9} + \frac{- 4}{x + 2}$