Simplify ((x^2-36)/(3x+18))÷((x^2+x-42)/(3x^2))

$\frac{x^{2} - 36}{3 x + 18} \div \frac{x^{2} + x - 42}{3 x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 36}{3 x + 18} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Step 2

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{3 x + 18} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{3 x + 18} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Step 3

Simplify terms.

Factor $3$ out of $3 x + 18$ .

Factor $3$ out of $3 x$ .

$\frac{(x + 6) (x - 6)}{3 (x) + 18} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Factor $3$ out of $18$ .

$\frac{(x + 6) (x - 6)}{3 x + 3 \cdot 6} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Factor $3$ out of $3 x + 3 \cdot 6$ .

$\frac{(x + 6) (x - 6)}{3 (x + 6)} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{(x + 6) (x - 6)}{3 (x + 6)} \cdot \frac{3 (x^{2})}{x^{2} + x - 42}$

Cancel the common factor.

$\frac{(x + 6) (x - 6)}{3 (x + 6)} \cdot \frac{3 x^{2}}{x^{2} + x - 42}$

Rewrite the expression.

$\frac{(x + 6) (x - 6)}{x + 6} \cdot \frac{x^{2}}{x^{2} + x - 42}$

Multiply $\frac{(x + 6) (x - 6)}{x + 6}$ by $\frac{x^{2}}{x^{2} + x - 42}$ .

$\frac{(x + 6) (x - 6) x^{2}}{(x + 6) (x^{2} + x - 42)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x + 6) (x - 6) x^{2}}{(x + 6) (x^{2} + x - 42)}$

Rewrite the expression.

$\frac{(x - 6) x^{2}}{x^{2} + x - 42}$

Step 4

Factor $x^{2} + x - 42$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 42$ and whose sum is $1$ .

$- 6, 7$

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{2}}{x + 7}$

Simplify ((x^2-36)/(3x+18))÷((x^2+x-42)/(3x^2))

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