Question

A roll of material is (x + 2) metres wide. Annie buys (x + 3) metres of the material and Bronwyn buys 5 metres of the material.

Answer 1

Annie and Bronwyn buy two rectangles of material.

The width is common and it is x+2.

The height of Annie's rectangle is x+3, the height of Bronwyn is 5.

Since the area of a rectangle is the product of width and height, Annie's rectangle has an area of

$A = (x+2)(x+3)$

And Bronwyn's rectangle has an area of

$B = 5(x+2)$

If Annie bought more than Bronwyn, the difference is

$A-B = (x+2)(x+3)-5(x+2) = (x+2)[(x+3)-5] = (x+2)(x-2)$

You can write this expression as you prefer: $(x+2)(x-2)=x^2-4$

Question (d) represents the situation

$A=B+5 \implies (x+2)(x+3)=5(x+2)+5$

We can manipulate and solve this equation as follows:

$(x+2)(x+3)=5(x+2)+5 \iff x^2+5x+6=5x+15 \iff x^2-9=0 \iff x = \pm 3$

We can't accept the solution x = -3 because it would lead to negative dimensions. So, we have x=3, which implies the following dimensions and area:

$A = (x+2)(x+3) = 5\cdot 6 = 30$

$B = 5(x+2) = 5\cdot 5 = 25$

which indeed means that Annie has 5 squared meters more material than Bronwyn.