Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Cancel the common factor of $3 x$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

$(9 - x^{2}) \frac{1}{x^{2} + 6 x + 9}$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$(9 - x^{2}) \frac{1}{x^{2} + 6 x + 3^{2}}$

Step 3.2

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 3.3

Rewrite the polynomial.

$(9 - x^{2}) \frac{1}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}$

Step 3.4

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

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

Step 4

Multiply $9 - x^{2}$ by $\frac{1}{{(x + 3)}^{2}}$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 6

Cancel the common factor of $3 + x$ and ${(x + 3)}^{2}$ .

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Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x + 3$ out of ${(x + 3)}^{2}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$\frac{3 - x}{x + 3}$