Question

This shows the template for a diamond-shaped mirror frame. (View attachment)

If Jeff cuts out the shaded area for the mirror, what is the area of the remaining wooden frame?
a) Write the expression for the area.
b) Then expand and simplify.

Answer 1

We know that the area of a square of side length L is given by:

A = L^2

Here, the total area of the white square is:

A = (3x + 1)^2

And the area of the shaded square is:

A' = (2x + 3)^2

a) If we remove the shaded area from the white area, the remaining area is just the difference between the two, then:

area = A - A' = (3x + 1)^2 - (2x + 3)^2

This is the expression we wanted.

b) Now we need to expand and simplify the expression:

area = (3x + 1)^2 - (2x + 3)^2

area = (3x)^2 + 2*(3x)*1 + 1^2 - (2x)^2 - 2*(2x)*3 - 3^2

area = 9x^2 + 6x + 1 - 4x^2 - 12x - 9

area = (9 - 4)*x^2 + (6 - 12)*x + 1 - 9

area = 5*x^2 - 6*x - 8

This is the simplified equation for the remaining area.