Question

I need help finding the LCD for this rato expression.

Answer 1

We are given the following expressions

$\frac{14}{3x^2-18x-48}\; and\; \frac{-1}{21x^2-84}$

We are asked to find the least common denominator (LCD) for the above expressions.

First of all, we need to factor out both the denominators

$\begin{gathered} 3x^2-18x-48 \\ 3(x^2-6x-16) \\ 3(x^2-8x+2x-16) \\ 3((x^2-8x)+(2x-16)) \\ 3(x(x^{}-8)+2(x-8)) \\ 3(x^{}-8)(x+2) \end{gathered}$

Similarly, factor out the other denominator

$\begin{gathered} 21x^2-84 \\ 21(x^2-4) \\ 21((x)^2-(2)^2) \\ 21(x+2)(x-2) \end{gathered}$

So, the expressions become

$\frac{14}{3(x^{}-8)(x+2)}\; and\; \frac{-1}{21(x+2)(x-2)}$

The least common denominator (LCD) is

$21(x-8)(x+2)(x-2)$