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:

`(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=(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=(-3\pm7)/(2) \\ x_1=(-3-7)/(2)=-(10)/(2)=-5 \\ x_2=(-3+7)/(2)=(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:

`(x^2-4x+4)/(x^2+3x-10)=((x-2)^2)/((x+5)(x-2))=(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.