Math Review of Difference of Squares

Overview

The difference of squares is a common binomial pattern that can be factored as (x+b)(x-b). It can be used in situations involving estimation by making both elements a difference of squares.

Perfect Squares

In order for the pattern to work, both elements must be perfect squares. The variable x² is a perfect square. So is 9x⁴, as it can be factored as (3x²)². Similarly, the numbers 36(6²) and 81(9²) are perfect squares. The monomial 100x² can be factored as 10x(10x).

Difference of Squares

Suppose the expression were 25x² – 144. Both 25x² and 144 are perfect squares. The monomial 25x² can be factored as 5x∙5x and 144 can be factored as 12∙12. The entire expression can be factored as (5x +12)(5x -12). The middle terms 5x∙12 and 5x∙-12 cancel each other out.

Figure 1: The pattern for the difference of squares shows that the middle terms cancel each other.

Using Estimation

A math problem such as 27∙33 can be estimated as the difference of squares. The mean of 27 and 33 is 30, so 27 times 33 can be expressed as (30 – 3)(30 + 3) or 30² – 3² or 900 – 9 which is 891. In order to check, 27∙33 is 891.  Similarly, 56∙44 can be expressed as (50 + 6)(50 – 6) or 2500 – 36 = 2464, and 56∙44 is 2464.

Figure 2: An example of estimation using the difference of squares.

Geometric Representation

Suppose one square has the area a² and one corner with the area measuring b² is taken out of it. The resulting figure, a²-b², can be divided into 2 rectangles. One rectangle has the measurement of a(a-b). The other rectangle has the measurement of b(a-b). Adding the two rectangles together gives the measurement of (a+b)(a-b) for the same figure. The area of the figure is (a +b)(a-b), so a²-b² = (a+b)(a-b).

Figure 3: Geometric representation of the difference of squares.

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