Question

How to work out the length and the width of the rectangle

Answer 1

Rectangle:

Define x:

Let the width be x.
width = x
Length = 3x + 4

Find Area:

Area = x(3x + 4)

Square:
Length of square = Length of rectangle = 3x + 4

Area = (3x + 4)²

Area of square is 66cm² more than area of rectangle.
⇒ (3x+ 4)² = x(3x + 4) + 66

Solve x;
(3x+ 4)² = x(3x + 4) + 66

Remove brackets:
9x² + 24x + 16 = 3x² + 4x + 66 

Take away 3x² + 4x + 66 from both sides:
6x² + 20x - 50 = 0

Divide by 2: 
3x² + 10x - 25 = 0

Factorise:
(3x - 5)(x + 5) = 0

Apply product multiplication property
(3x - 5)= 0 or (x + 5) = 0
x = 5/3 or x = -5 (rejected, since x cannot be negative)

Width x= 5/3 cm
Length = 3x + 4 = 3(5/3) + 4 = 9 cm

Answer: The width is 5/3 cm and length is 9 cm.