Algebra Examples

$\frac{x}{3 - x} + \frac{4 x + 7}{x^{2} - 9}$

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 3$ .

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{x}{3 - x} + \frac{4 x + 7}{(x + 3) (- 1 (- x) - 3)}$

Step 2.2

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

$\frac{x}{3 - x} + \frac{4 x + 7}{(x + 3) (- 1 (- x) - 1 (3))}$

Step 2.3

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

$\frac{x}{3 - x} + \frac{4 x + 7}{(x + 3) (- 1 (- x + 3))}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{4 x + 7}{(x + 3) (- 1 (- x + 3))}$ to the numerator.

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

Step 2.4.2

Reorder terms.

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

Step 3

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

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

Step 4

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 4.1

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

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

Step 4.2

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Move $3$ to the left of $x$ .

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

Step 6.4

Apply the distributive property.

$\frac{x^{2} + 3 x - (4 x) - 1 \cdot 7}{(- x + 3) (x + 3)}$

Step 6.5

Multiply $4$ by $- 1$ .

$\frac{x^{2} + 3 x - 4 x - 1 \cdot 7}{(- x + 3) (x + 3)}$

Step 6.6

Multiply $- 1$ by $7$ .

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

Step 6.7

Subtract $4 x$ from $3 x$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

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

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

Step 7.2

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

$\frac{x^{2} - x - 7}{(- (x) - 1 (- 3)) (x + 3)}$

Step 7.3

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

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

Step 7.4

Rewrite negatives.

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Step 7.4.1

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

$\frac{x^{2} - x - 7}{- 1 (x - 3) (x + 3)}$

Step 7.4.2

Move the negative in front of the fraction.

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