Q. 7.15 - Intermediate Algebra

Question

Simplify $\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}$.

Step-by-Step Solution

Answer

The simplest form of the given expression is$\frac{(x+2)}{4}.$

Step 1 . Given information

The given expression is$\frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}.$

Step 2 . Simplify

$$\begin{gathered} \frac{{\displaystyle \frac{3x^2+7x+2}{4x+24}}}{{\displaystyle \frac{3x^2-14x-5}{x^2+x-30}}}=\frac{3x^2+7x+2}{4x+24}·\frac{{\displaystyle x^2+x-30}}{{\displaystyle 3x^2-14x-5}} \text{Convert divistion into multiplication} \\ =\frac{(x+2)(3x+1)}{4(x+6)}·\frac{{\displaystyle (x+6)(x-5)}}{{\displaystyle (x-5)(3x+1)}} \text{Factorize} \\ =\frac{(x+2)(3x+1)(x+6)(x-5)}{4(x+6)(x-5)(3x+1)} \text{Multiply numerator and denominator} \\ =\frac{(x+2)}{4} \text{Cancel common factors of numerator and denominator} \end{gathered}$$

Hence, the given expresion is simplified as $\frac{(x+2)}{4}.$