QuestionQuestion 2
Simplify the following expression completely. $\frac{4 x^{2}+7 x+3}{4 x^{2}-9 x-9}$
Enter the numerator and denominator separately in the boxes below. If the denominator is.1, enter the number 1. Do not leave either box blank. Answer: $\square$ $\square$ Numerator preview: Denominator preview: Question Help: Message instructor

Studdy Solution

STEP 1

1. We are given a rational expression to simplify.
2. The expression can be simplified by factoring both the numerator and the denominator.
3. We will factor each polynomial separately and then simplify the expression by canceling common factors.

STEP 2

1. Factor the numerator $4x^2 + 7x + 3$ .
2. Factor the denominator $4x^2 - 9x - 9$ .
3. Simplify the expression by canceling common factors, if any.

STEP 3

Factor the numerator $4x^2 + 7x + 3$ .
We look for two numbers that multiply to $4 \times 3 = 12$ and add to $7$ . The numbers $3$ and $4$ satisfy this condition.
Rewrite the middle term using these numbers:
$4x^2 + 4x + 3x + 3$
Group the terms:
$(4x^2 + 4x) + (3x + 3)$
Factor by grouping:
$4x(x + 1) + 3(x + 1)$
Factor out the common binomial:
$(4x + 3)(x + 1)$
The factored form of the numerator is $(4x + 3)(x + 1)$ .

STEP 4

Factor the denominator $4x^2 - 9x - 9$ .
We look for two numbers that multiply to $4 \times (-9) = -36$ and add to $-9$ . The numbers $-12$ and $3$ satisfy this condition.
Rewrite the middle term using these numbers:
$4x^2 - 12x + 3x - 9$
Group the terms:
$(4x^2 - 12x) + (3x - 9)$
Factor by grouping:
$4x(x - 3) + 3(x - 3)$
Factor out the common binomial:
$(4x + 3)(x - 3)$
The factored form of the denominator is $(4x + 3)(x - 3)$ .

STEP 5

Simplify the expression by canceling the common factor $(4x + 3)$ .
The simplified expression is:
$\frac{(x + 1)}{(x - 3)}$
Numerator: $x + 1$
Denominator: $x - 3$

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