Precalculus Examples

$\frac{5 x}{2 x^{2} - 18} + \frac{4}{3 x - 9}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

Factor $2$ out of $2 x^{2} - 18$ .

Tap for more steps...

Step 1.1.1.1

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

$\frac{5 x}{2 (x^{2}) - 18} + \frac{4}{3 x - 9}$

Step 1.1.1.2

Factor $2$ out of $- 18$ .

$\frac{5 x}{2 x^{2} + 2 \cdot - 9} + \frac{4}{3 x - 9}$

Step 1.1.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 9$ .

$\frac{5 x}{2 (x^{2} - 9)} + \frac{4}{3 x - 9}$

Step 1.1.2

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

$\frac{5 x}{2 (x^{2} - 3^{2})} + \frac{4}{3 x - 9}$

Step 1.1.3

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 $b = 3$ .

$\frac{5 x}{2 (x + 3) (x - 3)} + \frac{4}{3 x - 9}$

Step 1.2

Factor $3$ out of $3 x - 9$ .

Tap for more steps...

Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{5 x}{2 (x + 3) (x - 3)} + \frac{4}{3 (x) - 9}$

Step 1.2.2

Factor $3$ out of $- 9$ .

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

Step 1.2.3

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

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

Step 2

To write $\frac{5 x}{2 (x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3

To write $\frac{4}{3 (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{2 (x + 3)}{2 (x + 3)}$ .

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

Step 4

Write each expression with a common denominator of $6 (x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{5 x}{2 (x + 3) (x - 3)}$ by $\frac{3}{3}$ .

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

Step 4.2

Multiply $3$ by $2$ .

$\frac{5 x \cdot 3}{6 (x + 3) (x - 3)} + \frac{4}{3 (x - 3)} \cdot \frac{2 (x + 3)}{2 (x + 3)}$

Step 4.3

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

$\frac{5 x \cdot 3}{6 (x + 3) (x - 3)} + \frac{4 (2 (x + 3))}{3 (x - 3) (2 (x + 3))}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{5 x \cdot 3}{6 (x + 3) (x - 3)} + \frac{4 (2 (x + 3))}{6 (x - 3) (x + 3)}$

Step 4.5

Reorder the factors of $6 (x - 3) (x + 3)$ .

$\frac{5 x \cdot 3}{6 (x + 3) (x - 3)} + \frac{4 (2 (x + 3))}{6 (x + 3) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 3 + 4 (2 (x + 3))}{6 (x + 3) (x - 3)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $3$ by $5$ .

$\frac{15 x + 4 \cdot 2 (x + 3)}{6 (x + 3) (x - 3)}$

Step 6.2

Multiply $4$ by $2$ .

$\frac{15 x + 8 (x + 3)}{6 (x + 3) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{15 x + 8 x + 8 \cdot 3}{6 (x + 3) (x - 3)}$

Step 6.4

Multiply $8$ by $3$ .

$\frac{15 x + 8 x + 24}{6 (x + 3) (x - 3)}$

Step 6.5

Add $15 x$ and $8 x$ .

$\frac{23 x + 24}{6 (x + 3) (x - 3)}$