Simplify ((3x^2-x-14)/(9x^2-4))÷((x+2)/(3x^2+2x))

$\frac{3 x^{2} - x - 14}{9 x^{2} - 4} \div \frac{x + 2}{3 x^{2} + 2 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - x - 14}{9 x^{2} - 4} \cdot \frac{3 x^{2} + 2 x}{x + 2}$

Step 2

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 14 = - 42$ and whose sum is $b = - 1$ .

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

$\frac{3 x^{2} - (x) - 14}{9 x^{2} - 4} \cdot \frac{3 x^{2} + 2 x}{x + 2}$

Rewrite $- 1$ as $6$ plus $- 7$

$\frac{3 x^{2} + (6 - 7) x - 14}{9 x^{2} - 4} \cdot \frac{3 x^{2} + 2 x}{x + 2}$

Apply the distributive property.

$\frac{3 x^{2} + 6 x - 7 x - 14}{9 x^{2} - 4} \cdot \frac{3 x^{2} + 2 x}{x + 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 6 x) - 7 x - 14}{9 x^{2} - 4} \cdot \frac{3 x^{2} + 2 x}{x + 2}$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

Step 3

Simplify the denominator.

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Rewrite $4$ as $2^{2}$ .

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

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

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

Step 4

Simplify terms.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 5

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

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

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

Factor $x$ out of $2 x$ .

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

Factor $x$ out of $x (3 x) + x \cdot 2$ .

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

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

Step 6

Cancel the common factor of $3 x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Reorder factors in $\frac{(3 x - 7) x}{3 x - 2}$ .

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

Simplify ((3x^2-x-14)/(9x^2-4))÷((x+2)/(3x^2+2x))

Related Questions

Support LCMGCF.com by disabling your adblocker.