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