Question

In terms of x, find an expression that represents the area of the shaded region. The outer square has side lengths of (x+5) and the inner square has side lengths of (x-2), as shown.

Answer 1

Area = 14x + 21 square units

Explanation:

The formula of an area of a square with side length a:

`A=a^2`

The big square:

`a=x+5`

Substitute:

`A_B=(x+5)^2` use
`(a+b)^2=a^2+2ab+b^2`

`A_B=x^2+2(x)(5)+5^2=x^2+10x+25`

The small square:

`a=x-2`

Substitute:

`A_S=(x-2)^2` use
`(a-b)^2=a^2-2ab+b^2`

`A_S=x^2-2(x)(2)+2^2=x^2-4x+4`

The area of a shaded region:

`A=A_B-A_S`

Substitute:

`A=(x^2+10x+25)-(x^2-4x+4)=x^2+10x+25-x^2+4x-4`

combine like terms

`A=(x^2-x^2)+(10x+4x)+(25-4)=14x+21`