Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{4 x - 3}{x^{2} - 3^{2}} - \frac{2 x - 3}{x - 3}$

Step 1.2

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 $b = 3$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

Multiply $- 1$ by $- 3$ .

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

Step 5.4

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

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Step 5.4.1

Apply the distributive property.

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

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

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

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Step 5.5.1.1.1

Move $x$ .

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

Step 5.5.1.1.2

Multiply $x$ by $x$ .

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

Step 5.5.1.2

Multiply $3$ by $- 2$ .

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

Step 5.5.1.3

Multiply $3$ by $3$ .

$\frac{4 x - 3 - 2 x^{2} - 6 x + 3 x + 9}{(x + 3) (x - 3)}$

Step 5.5.2

Add $- 6 x$ and $3 x$ .

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

Step 5.6

Subtract $3 x$ from $4 x$ .

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

Step 5.7

Add $- 3$ and $9$ .

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

Step 5.8

Reorder terms.

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

Step 5.9

Factor by grouping.

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Step 5.9.1

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 $a \cdot c = - 2 \cdot 6 = - 12$ and whose sum is $b = 1$ .

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Step 5.9.1.1

Multiply by $1$ .

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

Step 5.9.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 5.9.1.3

Apply the distributive property.

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

Step 5.9.2

Factor out the greatest common factor from each group.

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Step 5.9.2.1

Group the first two terms and the last two terms.

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

Step 5.9.2.2

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

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

Step 5.9.3

Factor the polynomial by factoring out the greatest common factor, $- 2 x - 3$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x$ .

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

Step 6.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 6.3

Factor $- 1$ out of $- (2 x) - 1 (3)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (2 x + 3)$ as $- 1 (2 x + 3)$ .

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

Step 6.4.2

Move the negative in front of the fraction.

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