Difference of Two Squares

The difference of two squares theorem states that a quadratic equation can be written as a product of two binomials, one showing the difference of the square roots and the other showing the sum of the square roots. It is an algebraic representation of the equation used to express the difference between two square values. This type of formula allows us to solve a mathematical equation of the difference of squares quickly.

In this article, we will learn about the difference of two squares theorems, and also we will learn how to write the suitable identity to evaluate 103 x 107.

Square Objects

Difference of Squares Formula

The formula for the difference between the squares of two numbers is given by:

Difference of Square Formula

Where the first and last terms are both perfect squares.

On expanding the above image equation, we get,

$(a+b)(a-b)=a^{2}-a b+a b-b^{2}$

$(a+b)(a-b)=a^{2}-b^{2}$

Using the Difference of Square Formula

First, consider the following questions:

What is the square of 9? The answer is 81. So, how about 6 squares? The answer is 36.

Do you notice how 81-36 is the same as $9^{2}-6^{2}$?

How about the equation $a^{2}-b^{2}=(a+b)(a-b)$ ?

Now, comparing the equation, we get 9 as the letter $a$ and 6 as the letter $b$.

For 9 and 6, let's apply the difference of squares formula:

$9^{2}-6^{2} =(9+6)(9-6)$

$=15 \times 3$

$=45$

Have you noticed we might simply have subtracted 36 from 81 to have the same result? But what if the number was $86^{2}-77^{2}$? Finding a squared value for each and performing subtraction is a time taking process. The quickest and the easiest way to the answer is to recognise a difference in squares and use the formula:

In this example, you can see $a=86$ and $b=77$.

Substituting these values in the difference of square formula, we get,

$86^{2}-77^{2} =(86+77)(86-77)$

$=163 \times 9$

$=1467$

Example: Use suitable identity to evaluate: 97 x 103?

In this example, we will use a suitable identity to evaluate: 97 x 103. The identity is $(a+b)(a-b)=a^{2}-b^{2}$.

$97 \times 103$ can be rewritten as $(100-3)(100+3)$

$\therefore 97 \times 103=(100)^{2}-(3)^{2}$

$=10000-9$

$=9991$

Examples: The difference of the squares of two numbers is 88. If the larger number is 5, less than twice the smaller number, then the product of the two numbers is:

Ans: Let smaller number be $x$ and larger number be $y$ According to the question, $\Rightarrow \mathrm{y}^{2}-\mathrm{x}^{2}=88$ $\Rightarrow \mathrm{y}=2 \mathrm{x}-5$

Substituting the value of $y$ from equation (2) in (1),

$(2 x-5)^{2}-x^{2}=88$

$\Rightarrow 4 \mathrm{x}^{2}-20 \mathrm{x}+25-\mathrm{x}^{2}-88=0$

$\Rightarrow 3 \mathrm{x}^{2}-20 \mathrm{x}-63=0$

$\Rightarrow 3 \mathrm{x}^{2}-27 \mathrm{x}+7 \mathrm{x}-63=0$ $\Rightarrow 3 \mathrm{x}(\mathrm{x}-9)+7(\mathrm{x}-9)=0$ $\Rightarrow(\mathrm{x}-9)(3 \mathrm{x}+7)=0$

$\Rightarrow \mathrm{x}-9=0$ and $3 \mathrm{x}+7=0$ $\Rightarrow \mathrm{x}=9$ and $\mathrm{x}=-\dfrac{7}{3}$

The number cannot be negative.

$\therefore$ Smaller number $=9$

$\Rightarrow$ Larger number $=2 \mathrm{x}-5=2 \times 9-5=18-5=13$

$\Rightarrow$ Product of two numbers $=9 \times 13=117$

Therefore, if the difference of the squares of two numbers is 88, then the product of two numbers is 117.

Solved Examples for the Difference in Squares

Q1. Evaluate $12^{2}-9^{2}$ using the difference of squares.

Ans: Remember that a difference of two squares has the form $a^{2}-b^{2}$ and that it may be factored using the formula $a^{2}-b^{2}=(a+b)(a-b)$.

On comparing the given expression with $a^{2}-b^{2}$, we have,

$a^{2}=12^{2} \Rightarrow a=12$

$b^{2}=9^{2} \Rightarrow b=9$

Putting the values in the difference of squares formula, we get;

$(12+9)(12-9)=(21)(3)=63$

Q2. Write the suitable identity to evaluate 103 x 107

Ans: In this question, we must write the suitable identity to evaluate 103 x 107. So for this, we will use the identity and then solve it.

The identity is: $(x+a)(x+b)=x^{2}+(a+b) x+a b$

To split $103 \times 107$, we need a square that is easy to calculate.

Hence, $x=100, a=3, b=7$

$\therefore 103 \times 107=(100+3)(100+7)$

$=100^{2}+(3+7)(100)+(3)(7)$

$=10000+1000+21=11021$

Factor Using the Difference of Square

We will learn how to factorise algebraic problems using the difference of squares formula in this portion. Recall that a difference of two squares has the form $a^{2}-b^{2}$ and that it may be factored using the formula $a^{2}-b^{2}=(a+b)(a-b)$.

The following procedures are taken to factor a difference of squares:

Step 1: Check for the greatest common factor (GCF) of the terms and factor it out.

Step 2: Determine the numbers that will yield the same results and use the difference of squares formula: $a^{2}-b^{2}=(a+b)(a-b)$ or $(a-b)(a+b)$

Let's use these two steps to solve a few examples.

Q1. Factorize $7 a^{2}-63 b^{2}$

Ans: Here 7 is the Greatest Common Factor(GCF) of the terms, So take it out.

$7 a^{2}-63 b^{2}=7\left(a^{2}-9 b^{2}\right)$

$=7\left[(a)^{2}-(3 b)^{2}\right]$

Now apply $a^{2}-b^{2}=(a+b)(a-b)$ to get; $=7(a+3 b)(a-3 b)$

Q2. Factorize $a^{2}-36$

Ans: Since $a^{2}-36$ can be written as $a^{2}-6^{2}$.

Applying identity $a^{2}-b^{2}=(a+b)(a-b)$ we get;

$a^{2}-6^{2}=(a+6)(a-6)$

Practice on Your Own

Q1. Evaluate the expression using the difference of squares formula $42^{2}-39^{2}$.

Ans: 243

Q2. The square difference of the two numbers is 88. What is the product of two numbers if the larger number is 5 less than twice the smaller number?

Ans: Numbers are 9 and 13, and the product is 117. And $(x+4)(x-4)$

Summary

The difference of squares is an algebraic representation of the equation used to express the difference between two square values, i.e. $a^{2}-b^{2}$. The quickest and easiest way to simplify the difference in squares. The factor of an algebraic expression can be determined using the difference of the square formula. In this article, we have also seen how to solve the question based on the difference of the squares of two numbers is 88. In the end, we have included practice questions to use these identities and understand how to solve them.