How Do I Calculate the Difference between Two Squared numbers?
By Robert O
The difference of the two squares formula is useful when factoring quadratic expressions. In general form, a difference of two squares is (a² – b²). How does that come about?
If you are to expand (a – b)(a + b), the result is the difference of squares. Let us see how we get there.
![\[(a-b)(a+b)=a^{2}+a b-b a-b^{2}=a^{2}-b^{2}\]](data:image/svg+xml;nitro-empty-id=MjUwOjEyMTc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzM2IDIyIiB3aWR0aD0iMzM2IiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
This formula is very helpful in factoring expressions. Provided you can recognize that an expression involves the difference of two squares, you can apply the formula directly without even stating it. Your examiner already knows it.
How to factor polynomials using the difference of two squares formula?
For the formula to work, there must be a subtraction (-) separating the two terms. Any other operation apart from subtraction is not applicable. For instance, (a² + b²) is not a difference between the two squares. It may be a sum of two squares, but that does not exist.
We will use examples to help us in understanding this concept.
Example 1:
Factor x² – 4
Solution
This example is the simplest form of a difference of two squares. The first step is to confirm if it is indeed a difference between the two squares. That may not be obvious in some cases. So, you may need to first pull out a common factor before proceeding with the identification. In this case, we have only 1 to factor out, which makes no difference.
After confirming that it is a difference between the two squares, we check the numbers that make up the squares. The first term is x² and the second term is 4. Find the square root of each term independently. In this case, our square roots are x and 2. Note that if you cannot find a definite square root of individual terms, then that is not a difference between two squares.
Applying the general formula:
![\[(a-b)(a+b)=a^{2}+a b-b a-b^{2}=a^{2}-b^{2}\]](data:image/svg+xml;nitro-empty-id=MjUwOjMzNDY=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzM2IDIyIiB3aWR0aD0iMzM2IiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
![\[x^{2}-4=(x-2)(x+2)\]](data:image/svg+xml;nitro-empty-id=MjUwOjM3ODU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTgxIDIyIiB3aWR0aD0iMTgxIiBoZWlnaHQ9IjIyIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 2:
Factor 2x² – 98
Solution
A quick look at the problem tells you that it is not a difference of two squares. Do not fall into this trap. There is a common factor to pull out, and that is 2.
![\[2 x^{2}-98=2\left(x^{2}-49\right)\]](data:image/svg+xml;nitro-empty-id=MjUwOjQ1Nzc=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTczIDIzIiB3aWR0aD0iMTczIiBoZWlnaHQ9IjIzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
The terms in the bracket form a difference of two squares.
![\[2 x^{2}-98=2\left(x^{2}-49\right)=2 x^{2}-98=2(x-7)(x+7)\]](data:image/svg+xml;nitro-empty-id=MjUwOjUxMTk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNDA1IDIzIiB3aWR0aD0iNDA1IiBoZWlnaHQ9IjIzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Example 3:
Factor 32 – 8x²
Solution
Does this look like a difference of two squares? Let us factor out 8 and see.
![\[32-8 x^{2}=8\left(4-x^{2}\right)\]](data:image/svg+xml;nitro-empty-id=MjUwOjU4MjU=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTY0IDIzIiB3aWR0aD0iMTY0IiBoZWlnaHQ9IjIzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
We now have a difference of two squares in between the braces.
![\[32-8 x^{2}=8\left(4-x^{2}\right)=8(2-x)(2+x)\]](data:image/svg+xml;nitro-empty-id=MjUwOjYzNTk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMzA3IDIzIiB3aWR0aD0iMzA3IiBoZWlnaHQ9IjIzIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg== "Rendered by QuickLaTeX.com")
Homework
Factor:
![\[x^{2}-\frac{1}{4}\]](data:image/svg+xml;nitro-empty-id=MjUwOjY4Nzk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNTAgMzYiIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[5 x^{2}-125\]](data:image/svg+xml;nitro-empty-id=MjUwOjczMTA=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNzUgMTciIHdpZHRoPSI3NSIgaGVpZ2h0PSIxNyIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
![\[1-\frac{4}{9} x^{2}\]](data:image/svg+xml;nitro-empty-id=MjUwOjc3NDk=-1;base64,PHN2ZyB2aWV3Qm94PSIwIDAgNjAgMzYiIHdpZHRoPSI2MCIgaGVpZ2h0PSIzNiIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj48L3N2Zz4= "Rendered by QuickLaTeX.com")
Remarks
The difference between the two squares formula makes it easier to factorize polynomials. This formula is also applicable in simplifying complex algebraic fractions plus many other applications. The good news is that there is nothing complicated about it, and you can easily derive it when you need to use it.
About the Author
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.