Question

a rectangular living room has an area of 12x^2-6x-18 the width of the living room is (2x-3) what is the length of the living room

Answer 1

To find the length of the living room given its area and width, we can use the formula for the area of a rectangle:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

We are given the area \( A = 12x^2 - 6x - 18 \) and the width \( W = 2x - 3 \).

We can set up the equation:

\[
12x^2 - 6x - 18 = \text{Length} \times (2x - 3)
\]

To find the length, we divide both sides of the equation by the width:

\[
\text{Length} = \frac{12x^2 - 6x - 18}{2x - 3}
\]

Now, we need to simplify the right-hand side. First, we can factor the numerator if possible. The expression \( 12x^2 - 6x - 18 \) can be factored as follows:

1. Factor out the greatest common factor (GCF) from the numerator:
\[
12x^2 - 6x - 18 = 6(2x^2 - x - 3)
\]

2. Now factor the quadratic \( 2x^2 - x - 3 \). We need two numbers that multiply to \( 2 \times -3 = -6 \) and add to \(-1\). These numbers are \( -3 \) and \( 2 \).

Rewrite the quadratic:
\[
2x^2 - 3x + 2x - 3 = (2x^2 - 3x) + (2x - 3) = x(2x - 3) + 1(2x - 3) = (2x - 3)(x + 1)
\]

Thus, we have:
\[
12x^2 - 6x - 18 = 6(2x - 3)(x + 1)
\]

Now substituting this back into our expression for length:
\[
\text{Length} = \frac{6(2x - 3)(x + 1)}{2x - 3}
\]

Assuming \( 2x - 3 \neq 0 \), we can simplify this:
\[
\text{Length} = 6(x + 1)
\]

Thus, the length of the living room is:
\[
\text{Length} = 6x + 6
\]

So, the length of the living room is \( \boxed{6x + 6} \).