Simplify ((3x^2-27x)/(2x^2+13x-7))÷((3x)/(4x^2-1))

$\frac{3 x^{2} - 27 x}{2 x^{2} + 13 x - 7} \div \frac{3 x}{4 x^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 27 x}{2 x^{2} + 13 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Step 2

Factor $3 x$ out of $3 x^{2} - 27 x$ .

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

$\frac{3 x (x) - 27 x}{2 x^{2} + 13 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Factor $3 x$ out of $- 27 x$ .

$\frac{3 x (x) + 3 x (- 9)}{2 x^{2} + 13 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Factor $3 x$ out of $3 x (x) + 3 x (- 9)$ .

$\frac{3 x (x - 9)}{2 x^{2} + 13 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Step 3

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 $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$\frac{3 x (x - 9)}{2 x^{2} + 13 (x) - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Rewrite $13$ as $- 1$ plus $14$

$\frac{3 x (x - 9)}{2 x^{2} + (- 1 + 14) x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Apply the distributive property.

$\frac{3 x (x - 9)}{2 x^{2} - 1 x + 14 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{3 x (x - 9)}{(2 x^{2} - 1 x) + 14 x - 7} \cdot \frac{4 x^{2} - 1}{3 x}$

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

$\frac{3 x (x - 9)}{x (2 x - 1) + 7 (2 x - 1)} \cdot \frac{4 x^{2} - 1}{3 x}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{3 x (x - 9)}{(2 x - 1) (x + 7)} \cdot \frac{4 x^{2} - 1}{3 x}$

Step 4

Simplify terms.

Cancel the common factor of $3 x$ .

Cancel the common factor.

$\frac{3 x (x - 9)}{(2 x - 1) (x + 7)} \cdot \frac{4 x^{2} - 1}{3 x}$

Rewrite the expression.

$\frac{x - 9}{(2 x - 1) (x + 7)} (4 x^{2} - 1)$

Multiply $\frac{x - 9}{(2 x - 1) (x + 7)}$ by $4 x^{2} - 1$ .

$\frac{(x - 9) (4 x^{2} - 1)}{(2 x - 1) (x + 7)}$

Step 5

Simplify the numerator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{(x - 9) ({(2 x)}^{2} - 1)}{(2 x - 1) (x + 7)}$

Rewrite $1$ as $1^{2}$ .

$\frac{(x - 9) ({(2 x)}^{2} - 1^{2})}{(2 x - 1) (x + 7)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

$\frac{(x - 9) (2 x + 1) (2 x - 1)}{(2 x - 1) (x + 7)}$

Step 6

Cancel the common factor of $2 x - 1$ .

Cancel the common factor.

$\frac{(x - 9) (2 x + 1) (2 x - 1)}{(2 x - 1) (x + 7)}$

Rewrite the expression.

$\frac{(x - 9) (2 x + 1)}{x + 7}$

Simplify ((3x^2-27x)/(2x^2+13x-7))÷((3x)/(4x^2-1))

Related Questions

Support LCMGCF.com by disabling your adblocker.