Question

1. use patterns and structure a rectangle has a width that is twice the length. if the area of the rectangle is represented by the expression $18x^2 + 48x + 32$, what expression represents the length of the rectangle? explain.

Explanation:

Step1: Factor out common constant

Factor 2 from the area expression:
$18x^2 + 48x + 32 = 2(9x^2 + 24x + 16)$

Step2: Factor quadratic trinomial

Factor the perfect square trinomial inside the parentheses:
$9x^2 + 24x + 16 = (3x + 4)^2$
So the area becomes $2(3x + 4)^2$

Step3: Relate to length-width relationship

Let length = $l$, width = $2l$. Area = $l \times 2l = 2l^2$. Set equal to factored area:
$2l^2 = 2(3x + 4)^2$

Step4: Solve for length $l$

Divide both sides by 2, then take square root:
$l^2 = (3x + 4)^2$
$l = 3x + 4$ (since length is positive)

Answer:

The expression representing the length is $\boldsymbol{3x + 4}$. The area is factored to $2(3x + 4)^2$, which matches the form $2l^2$ (where width is twice the length, so area = $l \times 2l = 2l^2$), so solving for $l$ gives $3x + 4$.