The rectangle shown has an area of 27 cm2 and is modeled by the equation A = 2w2 + 3w, where w is the width. What is the length of the rectangle?

Answer 1

First the Area statement can be factored to show a relationship between the length and width, A = length x width.

A = 2w² + 3w

A = w(2w + 3) we know from this that the length must be equal to twice the width plus 3.

This can also serve as a guide line in factoring for the Quadratic Equation set up below and a check of the value we will obtain for the length.

We can set up a quadratic equation, then factor to get the width, use the width to find the length in the area equation and or in the relationship above, (l = 2w +3).

Starting with what we are given, the A = 27 cm² = 2w² + 3w

Subtract the area from both sides of the equation

0 = 2w² + 3w - 27

Use factoring, with factors of 27 and 2 arranged such that they give a difference or sum of 3

Factors of 27 include 1, 3, 9, 27

Factors of 2 are 1, 2

We want to set up the factors to use a +9 with a -6, this combination gives a +3

Factor the equation

(2w + 9)(w - 3)

w = 3 only since it cannot be -9/2

Since the width, w, is 3 cm we can calculate the length by using the formula for the area of a rectangle.

A = length x width = l x w = 27 cm²

27 cm² = l x 3 cm or 3l cm

We can divide both sides of the equation by 3 cm to find l

27cm²/3 cm = l

9 cm = l

So the length is 9 cm

Finally checking 9 = 2(3) + 3 = 2w + 3 from the relationship given above.

I hope you find this useful and please a leave a message if you have any questions.

Answer 2

2w² + 3w - 27

(2w + 9)(w - 3)

3