QuestionSimplify the expression: $\frac{x^{2}-9 x+14}{x^{2}+7 x-18}$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is $\frac{x^{}-9 x+14}{x^{}+7 x-18}$ . We need to simplify this expression to its lowest terms3. This involves factoring the numerator and the denominator

STEP 2

First, let's factor the numerator $x^{2}-9 x+14$ . This can be factored into two binomials of the form $(x-a)(x-b)$ , where $a$ and $b$ are numbers such that $a \times b =14$ and $a + b = -9$ .

STEP 3

By inspection, we can see that $a =2$ and $b =7$ satisfy these conditions. So, we can write the numerator as $x^{2}-9 x+14 = (x-2)(x-7)$

STEP 4

Next, let's factor the denominator $x^{2}+7 x-18$ . This can also be factored into two binomials of the form $(x-c)(x-d)$ , where $c$ and $d$ are numbers such that $c \times d = -18$ and $c + d =7$ .

STEP 5

By inspection, we can see that $c = -2$ and $d =9$ satisfy these conditions. So, we can write the denominator as $x^{2}+7 x-18 = (x-2)(x+9)$

STEP 6

Now we can write the original expression in factored form $\frac{x^{2}-9 x+14}{x^{2}+ x-18} = \frac{(x-2)(x-)}{(x-2)(x+9)}$

STEP 7

We can see that $(x-2)$ is a common factor in the numerator and the denominator. We can cancel out this common factor to simplify the expression $\frac{(x-2)(x-7)}{(x-2)(x+9)} = \frac{x-7}{x+9}$ The rational expression in lowest terms is $\frac{x-7}{x+9}$ .

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