Question

1. a rectangular living room has an area of $2x^2 + 9x - 18$. the width of the living room is $(2x - 3)$. what is the perimeter of the living room?

a. $2x^2 + 11x - 21$

b. $x + 6$

c. $6x + 6$

d. $4x - 6$

e. $3x + 3$

Explanation:

Step1: Find the length

The area of a rectangle is \( A = \text{length} \times \text{width} \). Given \( A = 2x^2 + 9x - 18 \) and width \( = 2x - 3 \), we factor the area or divide by the width to find the length. Let's perform polynomial division or factor the quadratic. Factoring \( 2x^2 + 9x - 18 \): we need two numbers that multiply to \( 2 \times (-18) = -36 \) and add to \( 9 \). Those numbers are \( 12 \) and \( -3 \). So, \( 2x^2 + 12x - 3x - 18 = 2x(x + 6) - 3(x + 6) = (2x - 3)(x + 6) \). Thus, length \( = x + 6 \).

Step2: Calculate the perimeter

The perimeter of a rectangle is \( P = 2(\text{length} + \text{width}) \). Substitute length \( = x + 6 \) and width \( = 2x - 3 \):
\( P = 2[(x + 6) + (2x - 3)] \)
Simplify inside the brackets: \( x + 6 + 2x - 3 = 3x + 3 \)
Then multiply by 2: \( P = 2(3x + 3) = 6x + 6 \)

Answer:

C. \( 6x + 6 \)