Simplify the following rational expression:
$\frac{x^2+6x-16}{x^2-4x+4} = \frac{x+[?]}{x-[ ]}$
We are given the rational expression \(\frac{x^2+6x-16}{x^2-4x+4}\) and aim to simplify it into the form \(\frac{x+[?]}{x-[?]}\). Our strategy involves factoring both the numerator and the denominator and canceling any common factors.
First, we factor the numerator \(x^2+6x-16\). We look for two numbers whose product is \(-16\) and whose sum is \(6\). These numbers are \(8\) and \(-2\). Therefore, we can write the numerator as:
\[x^2+6x-16 = (x+8)(x-2)\]
Next, we factor the denominator \(x^2-4x+4\). We observe that this is a perfect square trinomial, since \((x-2)^2 = x^2 - 4x + 4\). Thus:
\[x^2-4x+4 = (x-2)(x-2)\]
Now we substitute these factored forms back into the original rational expression:
\[\frac{x^2+6x-16}{x^2-4x+4} = \frac{(x+8)(x-2)}{(x-2)(x-2)}\]
We can see that the factor \((x-2)\) is common to both the numerator and the denominator. Assuming \(x
eq 2\) (so that we are not dividing by zero), we can cancel the common factor \((x-2)\):
\[\frac{(x+8)(x-2)}{(x-2)(x-2)} = \frac{x+8}{x-2}\]
Comparing the simplified expression \(\frac{x+8}{x-2}\) to the target form \(\frac{x+[?]}{x-[?]}\), we identify the missing constants:
\[\frac{x+8}{x-2} = \frac{x+8}{x-2}\]
Thus, the first missing value is \(8\) and the second missing value is \(2\). The simplified expression is \(\frac{x+8}{x-2}\).
Simplifying rational expressions is a vital skill in algebra used whenever you need to model relationships between quantities, such as calculating average speeds, determining rates of work, or simplifying complex physics formulas. For example, if a machine's efficiency is described by the ratio \(\frac{x^2+6x-16}{x^2-4x+4}\) relative to a variable \(x\), simplifying it allows you to analyze its behavior much more easily: \(\frac{x+8}{x-2}\). This shows clearly how efficiency changes as \(x\) varies, helping engineers or analysts predict performance more accurately and efficiently.