Basic Math Examples

$\frac{x}{x^{2} - 9} - \frac{7}{x^{2} - 6 x + 9}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

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

$\frac{x}{x^{2} - 3^{2}} - \frac{7}{x^{2} - 6 x + 9}$

Step 1.1.2

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 $b = 3$ .

$\frac{x}{(x + 3) (x - 3)} - \frac{7}{x^{2} - 6 x + 9}$

Step 1.2

Factor using the perfect square rule.

Tap for more steps...

Step 1.2.1

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

$\frac{x}{(x + 3) (x - 3)} - \frac{7}{x^{2} - 6 x + 3^{2}}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 2

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

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

Step 3

To write $- \frac{7}{{(x - 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

Raise $x - 3$ to the power of $1$ .

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

Step 4.3

Raise $x - 3$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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

Step 4.7

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

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

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 7$ by $3$ .

$\frac{x^{2} - 3 x - 7 x - 21}{{(x - 3)}^{2} (x + 3)}$

Step 6.6

Subtract $7 x$ from $- 3 x$ .

$\frac{x^{2} - 10 x - 21}{{(x - 3)}^{2} (x + 3)}$