Question

A rectangle has twice the area of a square. The length of the rectangle is 12 cm greater and the width is 8 cm greater than a side of the square. Find the side of the square.

Answer 1

Given: Let the side of the square = s cm.

Then rectangle’s length = s + 12 cm and width = s + 8 cm.

Rectangle area = (s + 12)(s + 8) and this equals twice the square’s area = 2s².

Step-wise calculation:

1. Set up equation:

(s + 12)(s + 8) = 2s²

2. Expand left side:

s² + 20s + 96 = 2s²

3. Rearrange:

0 = 2s² – s² – 20s – 96

⇒ s² – 20s – 96

= 0

4. Solve quadratic using the quadratic formula or factoring:

Discriminant Δ = (–20)² – 4(1)(–96)

= 400 + 384

= 784

`sqrt(Δ) = 28`

`s = (20 ± 28)/2`

⇒ `s = (20 + 28)/2 = 24` or `s = (20 - 28)/2 = -4`

5. Reject the negative root (lengths must be positive).

The side of the square is 24 cm.