"Perform the multiplication or division and simplify. x+34x2−9÷x2+7x+122x2+7x−15" was answered by students like you

High School
Algebra
Algebra I
Quadratic function and equation

osula9a

Answered question

2021-12-14

Perform the multiplication or division and simplify.

\frac{x + 3}{4 x^{2} - 9} \div \frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}

Answer & Explanation

zesponderyd

Beginner2021-12-15Added 41 answers

Step 1
Simplify: $\frac{x + 3}{4 x^{2} - 9} \div \frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}$
Step 2
Inverting and changing to multiplication:
$\frac{x + 3}{4 x^{2} - 9} \div \frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}$
$= \frac{x + 3}{4 x^{2} - 9} \times \frac{2 x^{2} + 7 x - 15}{x^{2} + 7 x + 12}$
factorizing the quadratics:
$= \frac{x + 3}{{(2 x)}^{2} - 3^{2}} \times \frac{2 x^{2} + 10 x - 3 x - 15}{x^{2} + 4 x + 3 x + 12}$
$= \frac{x + 3}{(2 x - 3) (2 x + 3)} \times \frac{2 x (x + 5) - 3 (x + 5)}{x (x + 4) + 3 (x + 4)}$
$= \frac{x + 3}{(2 x - 3) (2 x + 3)} \times \frac{(2 x - 3) (x + 5)}{(x + 3) (x + 4)}$
Step 3
cancelling the same terms, we get:
$= \frac{(x + 3)}{(2 x - 3) (2 x + 3)} \times \frac{(2 x - 3) (x + 5)}{(x + 3) (x + 4)}$
$= \frac{x + 5}{(2 x + 3) (x + 4)}$

Lindsey Gamble

Beginner2021-12-16Added 38 answers

Step 1
Given:

\frac{x + 3}{4 x^{2} - 9} \div \frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}

Divide

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

\frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}

by multiplying

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

by the reciprocal of

\frac{x^{2} + 7 x + 12}{2 x^{2} + 7 x - 15}

\frac{(x + 3) (2 x^{2} + 7 x - 15)}{(4 x^{2} - 9) (x^{2} + 7 x + 12)}

Factor the expressions that are not already factored.

\frac{(2 x - 3) (x + 3) (x + 5)}{(2 x - 3) (x + 3) (x + 4) (2 x + 3)}

Cancel out

(2 x - 3) (x + 3)

in both numerator and denominator.

\frac{x + 5}{(x + 4) (2 x + 3)}

Expand the expression.

\frac{x + 5}{2 x^{2} + 11 x + 12}

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