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

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

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

- The addition of them form a binomial $a+b$.
- 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.

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

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

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.

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.

- $3p$ and $4q$ are the terms in both binomials in the multiplication.
- 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.

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

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

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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