# $a^2-b^2$ identity

## Formula

$a^2-b^2$ $\,=\,$ $(a+b)(a-b)$

### Introduction

Let’s denote two unknown quantities by two variables $a$ and $b$.

1. The variable $a$ can be added to another variable $b$, and the sum of them is written as $a+b$ in mathematics. The addition of variables $a$ and $b$ forms a binomial $a+b$.
2. Let’s assume that the variable $a$ is greater than the variable $b$. The variable $b$ can be subtracted from the variable $a$, and the subtraction of variable $b$ from the variable $a$ forms another binomial $a-b$.
3. The squares of both variables are written as $a^2$ and $b^2$ mathematically. The subtraction of $b$ square from $a$ square is written as an expression $a^2-b^2$ in algebra.

According to the difference of squares property, the difference between squares of any two quantities is equal to the product of the sum and difference of the quantities. The difference of squares property can be written algebraically in terms of $a$ and $b$ as follows.

$a^2-b^2$ $\,=\,$ $(a+b) \times (a-b)$

$\implies$ $a^2-b^2$ $\,=\,$ $(a+b).(a-b)$

Therefore, the $a$ square minus $b$ square is equal to $a$ plus $b$ times $a$ minus $b$.

$\,\,\,\therefore\,\,\,\,\,\,$ $a^2-b^2$ $\,=\,$ $(a+b)(a-b)$

#### Uses

The $a$ square minus $b$ square is used as a formula in different cases in mathematics.

##### Factorization

$\implies$ $a^2-b^2$ $\,=\,$ $(a+b).(a-b)$

It is used as a formula to factorize the expression in which the square of one quantity is subtracted from the square of another quantity in square form.

##### Simplification

$\implies$ $(a+b) \times (a-b)$ $\,=\,$ $a^2-b^2$

##### Verification

The $a$ square minus $b$ square formula should be verified before using it as a formula in mathematics. So, let’s take $a \,=\, 6$ and $b \,=\, 4$. Now, substitute them in both expressions of the $a$ square minus $b$ square formula to calculate their values.

$(1).\,\,$ $a^2-b^2$ $\,=\,$ $6^2-4^2$ $\,=\,$ $36-16$ $\,=\,$ $20$

$(2).\,\,$ $(a+b)(a-b)$ $\,=\,$ $(6+4)(6-4)$ $\,=\,$ $10 \times 2$ $\,=\,$ $20$

Now, compare the values of expressions on both sides of the equation and we can observe that

1. The value of $a$ square minus $b$ square is equal to $20$.
2. Tthe value of $a+b$ times $a-b$ is also equal to $20$.

It proves that the value of $a$ square minus $b$ square equal to the value of $a$ plus $b$ times $a$ minus $b$, and they are always equal for all values of $a$ and $b$. Hence, the $a$ square minus $b$ square equals to $a$ plus $b$ times $a$ minus $b$ is generally called an algebraic identity.

#### Proofs

There are two different standard methods to derive the $a$ square minus $b$ square algebraic identity in mathematics. Now, let’s learn how to prove $a^2-b^2$ formula in mathematics.

##### Algebraic Method

Learn how to prove the $a$ square minus $b$ square rule mathematically in algebraic approach.

##### Geometric Method

Learn how to derive the $a$ squared minus $b$ squared law mathematically in geometric method.

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