QuestionSimplify the expressions and find excluded values: 7. $\frac{3 x-9}{x^{2}-6 x+9}$ , 8. $\frac{4 x-8}{x^{2}-4 x+4}$ .

Studdy Solution

STEP 1

Assumptions1. We are given two rational expressions $\frac{3 x-9}{x^{}-6 x+9}$ and $\frac{4 x-8}{x^{}-4 x+4}$ . . We need to simplify these expressions and find the numbers that must be excluded from the domain.

STEP 2

Let's start with the first expression. We can simplify the numerator and the denominator by factoring.
$\frac{ x-9}{x^{2}-6 x+9} = \frac{(x-)}{(x-)^2}$

STEP 3

Now, we can simplify the expression by cancelling out the common factor $(x-3)$ from the numerator and the denominator.
$\frac{3(x-3)}{(x-3)^2} = \frac{3}{x-3}$

STEP 4

The denominator of a fraction cannot be zero, so we need to find the values of $x$ that make the denominator zero. $x-3 =0$

STEP 5

olving the equation $x-3 =0$ gives us the value of $x$ that must be excluded from the domain.
$x =3$

STEP 6

Now, let's move on to the second expression. We can simplify the numerator and the denominator by factoring.
$\frac{4 x-8}{x^{2}-4 x+4} = \frac{4(x-2)}{(x-2)^2}$

STEP 7

Now, we can simplify the expression by cancelling out the common factor $(x-2)$ from the numerator and the denominator.
$\frac{4(x-2)}{(x-2)^2} = \frac{4}{x-2}$

STEP 8

The denominator of a fraction cannot be zero, so we need to find the values of $x$ that make the denominator zero. $x-2 =0$

STEP 9

olving the equation $x-2 =$ gives us the value of $x$ that must be excluded from the domain.
$x =2$ So, the simplified expressions are $\frac{3}{x-3}$ and $\frac{4}{x-2}$ , and the numbers that must be excluded from the domain are $3$ and $2$ respectively.

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