Precalculus Examples

$\frac{x}{3 x - 12} + \frac{4}{x^{2} - 16} = \frac{1}{3}$

Step 1

Factor each term.

Tap for more steps...

Step 1.1

Factor $3$ out of $3 x - 12$ .

Tap for more steps...

Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{x}{3 (x) - 12} + \frac{4}{x^{2} - 16} = \frac{1}{3}$

Step 1.1.2

Factor $3$ out of $- 12$ .

$\frac{x}{3 x + 3 \cdot - 4} + \frac{4}{x^{2} - 16} = \frac{1}{3}$

Step 1.1.3

Factor $3$ out of $3 x + 3 \cdot - 4$ .

$\frac{x}{3 (x - 4)} + \frac{4}{x^{2} - 16} = \frac{1}{3}$

Step 1.2

Rewrite $16$ as $4^{2}$ .

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 (x - 4), (x + 4) (x - 4), 3$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.6

The LCM of $3, 1, 3$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 2.7

The factor for $x - 4$ is $x - 4$ itself.

$(x - 4) = x - 4$

$(x - 4)$ occurs $1$ time.

Step 2.8

The factor for $x + 4$ is $x + 4$ itself.

$(x + 4) = x + 4$

$(x + 4)$ occurs $1$ time.

Step 2.9

The factor for $x - 4$ is $x - 4$ itself.

$(x - 4) = x - 4$

$(x - 4)$ occurs $1$ time.

Step 2.10

The LCM of $x - 4, x + 4, x - 4$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - 4) (x + 4)$

Step 2.11

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$3 (x - 4) (x + 4)$

Step 3

Multiply each term in $\frac{x}{3 (x - 4)} + \frac{4}{(x + 4) (x - 4)} = \frac{1}{3}$ by $3 (x - 4) (x + 4)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\frac{x}{3 (x - 4)} + \frac{4}{(x + 4) (x - 4)} = \frac{1}{3}$ by $3 (x - 4) (x + 4)$ .

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

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.2.1

Cancel the common factor.

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

Step 3.2.1.2.2

Rewrite the expression.

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

Step 3.2.1.3

Cancel the common factor of $x - 4$ .

Tap for more steps...

Step 3.2.1.3.1

Cancel the common factor.

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

Step 3.2.1.3.2

Rewrite the expression.

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

Step 3.2.1.4

Apply the distributive property.

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

Step 3.2.1.5

Multiply $x$ by $x$ .

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

Step 3.2.1.6

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

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

Step 3.2.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.8

Multiply $3 \frac{4}{(x + 4) (x - 4)}$ .

Tap for more steps...

Step 3.2.1.8.1

Combine $3$ and $\frac{4}{(x + 4) (x - 4)}$ .

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

Step 3.2.1.8.2

Multiply $3$ by $4$ .

$x^{2} + 4 x + \frac{12}{(x + 4) (x - 4)} ((x - 4) (x + 4)) = \frac{1}{3} (3 (x - 4) (x + 4))$

Step 3.2.1.9

Cancel the common factor of $(x - 4) (x + 4)$ .

Tap for more steps...

Step 3.2.1.9.1

Factor $(x - 4) (x + 4)$ out of $(x + 4) (x - 4)$ .

$x^{2} + 4 x + \frac{12}{(x - 4) (x + 4) (1)} ((x - 4) (x + 4)) = \frac{1}{3} (3 (x - 4) (x + 4))$

Step 3.2.1.9.2

Cancel the common factor.

$x^{2} + 4 x + \frac{12}{(x - 4) (x + 4) \cdot 1} ((x - 4) (x + 4)) = \frac{1}{3} (3 (x - 4) (x + 4))$

Step 3.2.1.9.3

Rewrite the expression.

$x^{2} + 4 x + 12 = \frac{1}{3} (3 (x - 4) (x + 4))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.1.1

Factor $3$ out of $3 (x - 4) (x + 4)$ .

$x^{2} + 4 x + 12 = \frac{1}{3} (3 ((x - 4) (x + 4)))$

Step 3.3.1.2

Cancel the common factor.

$x^{2} + 4 x + 12 = \frac{1}{3} (3 ((x - 4) (x + 4)))$

Step 3.3.1.3

Rewrite the expression.

$x^{2} + 4 x + 12 = (x - 4) (x + 4)$

Step 3.3.2

Expand $(x - 4) (x + 4)$ using the FOIL Method.

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

$x^{2} + 4 x + 12 = x (x + 4) - 4 (x + 4)$

Step 3.3.2.2

Apply the distributive property.

$x^{2} + 4 x + 12 = x \cdot x + x \cdot 4 - 4 (x + 4)$

Step 3.3.2.3

Apply the distributive property.

$x^{2} + 4 x + 12 = x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$

Step 3.3.3

Simplify terms.

Tap for more steps...

Step 3.3.3.1

Combine the opposite terms in $x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$ .

Tap for more steps...

Step 3.3.3.1.1

Reorder the factors in the terms $x \cdot 4$ and $- 4 x$ .

$x^{2} + 4 x + 12 = x \cdot x + 4 x - 4 x - 4 \cdot 4$

Step 3.3.3.1.2

Subtract $4 x$ from $4 x$ .

$x^{2} + 4 x + 12 = x \cdot x + 0 - 4 \cdot 4$

Step 3.3.3.1.3

Add $x \cdot x$ and $0$ .

$x^{2} + 4 x + 12 = x \cdot x - 4 \cdot 4$

Step 3.3.3.2

Simplify each term.

Tap for more steps...

Step 3.3.3.2.1

Multiply $x$ by $x$ .

$x^{2} + 4 x + 12 = x^{2} - 4 \cdot 4$

Step 3.3.3.2.2

Multiply $- 4$ by $4$ .

$x^{2} + 4 x + 12 = x^{2} - 16$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 4.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 4 x + 12 - x^{2} = - 16$

Step 4.1.2

Combine the opposite terms in $x^{2} + 4 x + 12 - x^{2}$ .

Tap for more steps...

Step 4.1.2.1

Subtract $x^{2}$ from $x^{2}$ .

$4 x + 12 + 0 = - 16$

Step 4.1.2.2

Add $4 x + 12$ and $0$ .

$4 x + 12 = - 16$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $12$ from both sides of the equation.

$4 x = - 16 - 12$

Step 4.2.2

Subtract $12$ from $- 16$ .

$4 x = - 28$

Step 4.3

Divide each term in $4 x = - 28$ by $4$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $4 x = - 28$ by $4$ .

$\frac{4 x}{4} = \frac{- 28}{4}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 28}{4}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28}{4}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $- 28$ by $4$ .

$x = - 7$