What is a Perfect Square Formula?
A perfect square is a number with an integer as its square root. However in algebra, when we need to calculate the square of any binomial expression, one with two variables say ‘a’ and ‘b’, we use the perfect square formula.
It can be used to factorize or to determine the square of the sum or difference of two terms. The following are two variations of the perfect square formula in algebra, with the main difference being the signs in the expressions that have ‘a’ and ‘b’ as variables:
\({(a+b)}^2=a^2+2ab+b^2\)
\({(a-b)}^2=a^2-2ab+b^2\)
Rapid Recall
Solved Examples
Example 1:
Simplify \({(5x+y)}^2\) using the perfect square formula.
Solution:
We have to find the value of \({(5x+y)}^2.\)
By using the perfect square formula, we can find the square of the given binomial.
Let’s consider \(a=5x\) and \(b=y.\)
Then,
\({(a+b)}^2=a^2+2ab+b^2\)
\({(5x+y)}^2={(5x)}^2+2(5x)(y)+{(y)}^2\)
\(=25x^2+10xy+y^2\)
Therefore, the square of the given equation \({(5x+y)}^2\) is \(25x^2+10xy+y^2.\)
Example 2:
Find the square of difference of two terms for \({(3x-2y)}^2.\)
Solution:
We have to find the value of \({(3x-2y)}^2.\)
By using the perfect square formula, we can find the square of difference of two terms in the given equation.
Then, \(a=3x\) and \(b=2y.\)
\({(a-b)}^2=a^2-2ab+b^2\)
\({(3x-2y)}^2={(3x)}^2-2(3x)(2y)+{(2y)}^2\)
\(={9x}^2-12xy+{4y}^2\)
Therefore, the square of \({(3x-2y)}^2\) is \({9x}^2-12xy+{4y}^2.\)
Example 3:
Determine the value of \({(24)}^2\) using the perfect square formula.
Solution:
We have to find the value of \({(24)}^2.\)
We can represent the number 24 as 25 – 1. By doing so, we can apply the perfect square formula to find its square value.
Thus,
\(a=25\) and \(b=1\)
Now,
\({(a-b)}^2=a^2-2ab+b^2\) [Apply the formula]
\(={(25)}^2-2(25)(1)+{(1)}^2\) [Substitute the value]
\(=625-50+1\)
\(=576\)
As a result, \({(24)}^2\) is 576.
We use the perfect square formula for various purposes like finance management, finding lengths and distances, calculating the height of falling objects and so on.
The perfect square formula in two variations we have in math or specifically in algebra are, (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².
A perfect square formula is a mathematical expression that is used to find the square of the sum or subtraction of two terms.