Question

Step 1 of 2: Reduce the rational expression to its lowest terms. x^2 - 4x + 4/×^2 + 3x - 10Step 2 of 2: Find the restricted values of X, if any, for the given rational expression.

Answer 1

We have the expression:

$\frac{x^2-4x+4}{x^2+3x-10}$

We have to factorize both numerator and denominator:

$\begin{gathered} \text{For }x^2-4x+4\colon \\ x=\frac{-(-4)\pm\sqrt[]{(-4)^2-4\cdot1\cdot4}}{2\cdot1} \\ x=\frac{4\pm\sqrt[]{16-16}}{2} \\ x=\frac{4\pm0}{2} \\ x=2 \\ \longrightarrow x^2-4x+4=(x-2)^2 \end{gathered}$$\begin{gathered} \text{For }x^2+3x-10\colon \\ x=\frac{-3\pm\sqrt[]{3^2-4\cdot1\cdot(-10)}}{2\cdot1} \\ x=\frac{-3\pm\sqrt[]{9+40}}{2} \\ x=\frac{-3\pm\sqrt[]{49}}{2} \\ x=\frac{-3\pm7}{2} \\ x_1=\frac{-3-7}{2}=-\frac{10}{2}=-5 \\ x_2=\frac{-3+7}{2}=\frac{4}{2}=2 \\ \to x^2+3x-10=(x+5)(x-2) \end{gathered}$

As we have a common factor, we can simplify the expression as:

$\frac{x^2-4x+4}{x^2+3x-10}=\frac{(x-2)^2}{(x+5)(x-2)}=\frac{x-2}{x+5}$

The restricted values for x are the ones that make the expression become undefined. This happens when, for example, the denominator becomes 0.

In this case, when x=-5, the denominator x+5=-5+5=0 and the expression is undefined.

This is the only restricted value for this expression.

Answer:

The reduced expression is (x-2)/(x+5).

The restricted value is x=-5.