Simplify (2x^2+13x+15)/(4x^2-9)*(2x^2+3x-9)/(x^2+2x-15)

$\frac{2 x^{2} + 13 x + 15}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Step 1

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 $a \cdot c = 2 \cdot 15 = 30$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$\frac{2 x^{2} + 13 (x) + 15}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Rewrite $13$ as $3$ plus $10$

$\frac{2 x^{2} + (3 + 10) x + 15}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Apply the distributive property.

$\frac{2 x^{2} + 3 x + 10 x + 15}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 3 x) + 10 x + 15}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

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

$\frac{x (2 x + 3) + 5 (2 x + 3)}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$\frac{(2 x + 3) (x + 5)}{4 x^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

Step 2

Simplify the denominator.

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

$\frac{(2 x + 3) (x + 5)}{{(2 x)}^{2} - 9} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

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

$\frac{(2 x + 3) (x + 5)}{{(2 x)}^{2} - 3^{2}} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

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 $b = 3$ .

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + 3 x - 9}{x^{2} + 2 x - 15}$

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 $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + 3 (x) - 9}{x^{2} + 2 x - 15}$

Rewrite $3$ as $- 3$ plus $6$

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} + (- 3 + 6) x - 9}{x^{2} + 2 x - 15}$

Apply the distributive property.

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{2 x^{2} - 3 x + 6 x - 9}{x^{2} + 2 x - 15}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x^{2} - 3 x) + 6 x - 9}{x^{2} + 2 x - 15}$

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

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{x (2 x - 3) + 3 (2 x - 3)}{x^{2} + 2 x - 15}$

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{x^{2} + 2 x - 15}$

Step 4

Factor $x^{2} + 2 x - 15$ 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

$\frac{(2 x + 3) (x + 5)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{(x - 3) (x + 5)}$

Step 5

Simplify terms.

Cancel the common factor of $x + 5$ .

Factor $x + 5$ out of $(2 x + 3) (x + 5)$ .

$\frac{(x + 5) (2 x + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{(x - 3) (x + 5)}$

Factor $x + 5$ out of $(x - 3) (x + 5)$ .

$\frac{(x + 5) (2 x + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{(x + 5) (x - 3)}$

Cancel the common factor.

$\frac{(x + 5) (2 x + 3)}{(2 x + 3) (2 x - 3)} \cdot \frac{(2 x - 3) (x + 3)}{(x + 5) (x - 3)}$

Rewrite the expression.

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

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

Factor $2 x - 3$ out of $(2 x + 3) (2 x - 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x + 3}{2 x + 3} \cdot \frac{x + 3}{x - 3}$

Multiply $\frac{2 x + 3}{2 x + 3}$ by $\frac{x + 3}{x - 3}$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (2x^2+13x+15)/(4x^2-9)*(2x^2+3x-9)/(x^2+2x-15)

Related Questions

Support LCMGCF.com by disabling your adblocker.