Question

how would I simplify the rational expression x^2_2x-24÷ x^2+7x+12?

Answer 1

we have the expression

$\frac{x^2-2x-24}{x^2+7x+12}$

step 1

Simplify the numerator

$x^2-2x-24$

Solve the quadratic equation, using the formula

a=1

b=-2

c=-24

substitute

$x=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(1)(-24)}}{2(1)}$$\begin{gathered} x=\frac{2\pm\sqrt[]{100}}{2} \\ \\ x=\frac{2\pm10}{2} \end{gathered}$

the values of x are

x=6 and x=-4

therefore

$x^2-2x-24=(x+4)(x-6)$

step 2

Simplify the denominator

$x^2+7x+12$

Solve the quadratic equation, using the formula

a=1

b=7

c=12

substitute

$x=\frac{-7\pm\sqrt[]{7^2-4(1)(12)}}{2(1)}$$x=\frac{-7\pm1}{2}$

the values of x are

x=-3 and x=-4

therefore

$x^2+7x+12=(x+3)(x+4)$

step 3

substitute the given values in the original expression

$\frac{(x+4)(x-6)}{(x+3)(x+4)}$

simplify

$\frac{x-6}{x+3}$