# $(a+b)(a-b)$ formula

## Formula

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

It is read as the $a$ plus $b$ times $a$ minus $b$ is equal to the $a$ squared minus $b$ squared.

### Introduction

Let the two literals $a$ and $b$ be two variables.

1. The addition of them form a binomial $a+b$.
2. The subtraction of them form another binomial $a-b$.

The two binomials are sum and difference basis binomials. The involvement of these two binomials in multiplication is a special case in mathematics. Hence, the product of them is generally called as the special product of binomials or the special binomial product in algebra.

$(a+b) \times (a-b)$

The product of the binomials $a+b$ and $a-b$ is simply written in the following form in mathematics.

$\implies$ $(a+b)(a-b)$

The special product of the binomials $a+b$ and $a-b$ is equal to the difference of squares of the terms $a$ and $b$.

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

The mathematical equation expresses that the product of sum and difference basis binomials, which contain the same terms is equal to the difference of them.

It is used as a formula in mathematics. Hence, it is called an algebraic identity.

#### Usage

It is mainly used as a formula to simplify an expression when the expression satisfies the following conditions.

1. The two binomials should be formed by the two variables.
2. The two variables should be connected with opposite signs in the binomials.

#### Verification

Let’s check this mathematical equation by taking some values for verification. Take $a = 7$ and $b = 3$, and find the values of both sides of the equation.

$(1). \,\,\,$ ${(a+b)}{(a-b)}$ $\,=\,$ ${(7+3)}{(7-3)}$ $\,=\,$ $10 \times 4$ $\,=\,$ $40$

$(2). \,\,\,$ $a^2-b^2$ $\,=\,$ ${(7)}^2-{(3)}^2$ $\,=\,$ $49-9$ $\,=\,$ $40$

$\,\,\, \therefore \,\,\,\,\,\,$ ${(7+3)}{(7-3)}$ $\,=\,$ ${(7)}^2-{(3)}^2$ $\,=\,$ $40$

Therefore, we can use this mathematical equation as an algebraic identity in mathematics.

#### Example

Simplify $(3p+4q)(3p-4q)$

Let us check the given expression to know whether we can use $a$ plus $b$ times $a$ minus $b$ formula or not.

1. $3p$ and $4q$ are the terms in both binomials in the multiplication.
2. The terms $3p$ and $4q$ are connected by the opposite signs plus and minus in the factors of the multiplication.

The two conditions of $(a+b)(a-b)$ formula are satisfied. So, we can use the formula for simplifying given algebraic expression.

Take $a = 3p$ and $b = 4q$ and substitute them in the $a+b$ times $a-b$ formula.

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

$\implies$ $(3p+4q)(3p-4q)$ $\,=\,$ $(3p)^2-(4q)^2$

$\,\,\,\therefore\,\,\,\,\,\,$ $(3p+4q)(3p-4q)$ $\,=\,$ $9p^2-16q^2$

Thus, we use the $(a+b)(a-b)$ special binomial product rule in mathematics.

#### Proofs

There are two possible ways to derive the $a$ plus $b$ times $a$ minus $b$ formula in mathematics.

##### Algebraic method

Learn how to prove the $a$ plus $b$ times $a$ minus $b$ algebraic identity by the multiplication of algebraic expressions.

##### Geometric method

Learn how to prove the $a$ plus $b$ times $a$ minus $b$ algebraic law geometrically from the areas of geometric shapes.

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