Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 3$ .

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

Step 2

To write $\frac{2 x - 1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2 x - 1}{x + 3}$ by $\frac{x - 3}{x - 3}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Expand $(2 x - 1) (x - 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Move $x$ .

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

Step 4.2.1.1.2

Multiply $x$ by $x$ .

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

Step 4.2.1.2

Multiply $- 3$ by $2$ .

$\frac{3 x - 4 + 2 x^{2} - 6 x - 1 x - 1 \cdot - 3}{(x + 3) (x - 3)}$

Step 4.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{3 x - 4 + 2 x^{2} - 6 x - x - 1 \cdot - 3}{(x + 3) (x - 3)}$

Step 4.2.1.4

Multiply $- 1$ by $- 3$ .

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

Step 4.2.2

Subtract $x$ from $- 6 x$ .

$\frac{3 x - 4 + 2 x^{2} - 7 x + 3}{(x + 3) (x - 3)}$

Step 4.3

Subtract $7 x$ from $3 x$ .

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

Step 4.4

Add $- 4$ and $3$ .

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

Step 4.5

Reorder terms.

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