Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box. Also, specify any restrictions on the variable.12² - 27/6x² +33x + 36Rational expression in lowest terms:Variable restrictions for the original expression: x

Answer 1

Explanation:

The expression is given below as

$\frac{12x^2-27}{6x^2+33x+36}$

Step 1:

Simplify both the numerator and denominator

$\begin{gathered} \frac{12x^{2}-27}{6x^{2}+33x+36} \\ 12x^2-27=3(4x^2-9)=3(2x-3)(2x+3) \\ 6x^2+33x+36=3(2x^2+11x+12) \\ 3(2x^2+11x+12)=3(2x^2+8x+3x+12) \\ 3(2x^2+11x+12)=3(2x(x+4)+3(x+4) \\ 3(2x^2+11x+12)=3(2x+3)(x+4) \end{gathered}$

By rewritng the expression , we will have

$\begin{gathered} \frac{12x^{2}-27}{6x^{2}+33x+36}=\frac{3(2x+3)(2x-3)}{3(x+4)(2x+3)} \\ \frac{12x^{2}-27}{6x^{2}+33x+36}=\frac{2x-3}{x+4} \end{gathered}$

Hence,

The simplified expression will be

$\frac{2x-3}{x+4}$

The variable restriction of the original expression will be

$\begin{gathered} 3(2x+3)(x+4)=0 \\ 2x+3=0,x+4=0 \\ 2x=-3,x=-4 \\ x=-\frac{3}{2},x=-4 \end{gathered}$

Hence,

The variable restriction for the original expression will be

$x\ne-\frac{3}{2},-4$