Algebra Examples

$\frac{x - 1}{4 x^{2} - 9} - \frac{x + 3}{2 x - 3}$

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{x - 1}{{(2 x)}^{2} - 9} - \frac{x + 3}{2 x - 3}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

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

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 $a = 2 x$ and $b = 3$ .

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

Step 2

To write $- \frac{x + 3}{2 x - 3}$ as a fraction with a common denominator, multiply by $\frac{2 x + 3}{2 x + 3}$ .

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

Step 3

Write each expression with a common denominator of $(2 x + 3) (2 x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{x + 3}{2 x - 3}$ by $\frac{2 x + 3}{2 x + 3}$ .

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

Step 3.2

Reorder the factors of $(2 x - 3) (2 x + 3)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $3$ .

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

Step 5.3

Expand $(- x - 3) (2 x + 3)$ using the FOIL Method.

Tap for more steps...

Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.4

Simplify and combine like terms.

Tap for more steps...

Step 5.4.1

Simplify each term.

Tap for more steps...

Step 5.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.4.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.4.1.2.1

Move $x$ .

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

Step 5.4.1.2.2

Multiply $x$ by $x$ .

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

Step 5.4.1.3

Multiply $- 1$ by $2$ .

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

Step 5.4.1.4

Multiply $3$ by $- 1$ .

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

Step 5.4.1.5

Multiply $2$ by $- 3$ .

$\frac{x - 1 - 2 x^{2} - 3 x - 6 x - 3 \cdot 3}{(2 x + 3) (2 x - 3)}$

Step 5.4.1.6

Multiply $- 3$ by $3$ .

$\frac{x - 1 - 2 x^{2} - 3 x - 6 x - 9}{(2 x + 3) (2 x - 3)}$

Step 5.4.2

Subtract $6 x$ from $- 3 x$ .

$\frac{x - 1 - 2 x^{2} - 9 x - 9}{(2 x + 3) (2 x - 3)}$

Step 5.5

Subtract $9 x$ from $x$ .

$\frac{- 8 x - 1 - 2 x^{2} - 9}{(2 x + 3) (2 x - 3)}$

Step 5.6

Subtract $9$ from $- 1$ .

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

Step 5.7

Reorder terms.

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

Step 5.8

Factor $2$ out of $- 2 x^{2} - 8 x - 10$ .

Tap for more steps...

Step 5.8.1

Factor $2$ out of $- 2 x^{2}$ .

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

Step 5.8.2

Factor $2$ out of $- 8 x$ .

$\frac{2 (- x^{2}) + 2 (- 4 x) - 10}{(2 x + 3) (2 x - 3)}$

Step 5.8.3

Factor $2$ out of $- 10$ .

$\frac{2 (- x^{2}) + 2 (- 4 x) + 2 (- 5)}{(2 x + 3) (2 x - 3)}$

Step 5.8.4

Factor $2$ out of $2 (- x^{2}) + 2 (- 4 x)$ .

$\frac{2 (- x^{2} - 4 x) + 2 (- 5)}{(2 x + 3) (2 x - 3)}$

Step 5.8.5

Factor $2$ out of $2 (- x^{2} - 4 x) + 2 (- 5)$ .

$\frac{2 (- x^{2} - 4 x - 5)}{(2 x + 3) (2 x - 3)}$

Step 6

Simplify with factoring out.

Tap for more steps...

Step 6.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{2 (- (x^{2}) - 4 x - 5)}{(2 x + 3) (2 x - 3)}$

Step 6.2

Factor $- 1$ out of $- 4 x$ .

$\frac{2 (- (x^{2}) - (4 x) - 5)}{(2 x + 3) (2 x - 3)}$

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

$\frac{2 (- (x^{2} + 4 x) - 5)}{(2 x + 3) (2 x - 3)}$

Step 6.4

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{2 (- (x^{2} + 4 x) - 1 (5))}{(2 x + 3) (2 x - 3)}$

Step 6.5

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (5)$ .

$\frac{2 (- (x^{2} + 4 x + 5))}{(2 x + 3) (2 x - 3)}$

Step 6.6

Simplify the expression.

Tap for more steps...

Step 6.6.1

Rewrite $- (x^{2} + 4 x + 5)$ as $- 1 (x^{2} + 4 x + 5)$ .

$\frac{2 (- 1 (x^{2} + 4 x + 5))}{(2 x + 3) (2 x - 3)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{2 (x^{2} + 4 x + 5)}{(2 x + 3) (2 x - 3)}$