How to prove the $a^2-b^2$ formula in Algebraic Method

Formula

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

The algebraic expression $a^2-b^2$ represents the difference of the two square quantities. It can be expressed in factoring form as the product of two special binomials $a+b$ and $a-b$. The factoring form of the difference of squares can be derived in mathematics algebraically according to factorization.

Difference of squares in Algebraic form

$a$ and $b$ represent two terms and the difference of squares of them is written as $a^2-b^2$ in mathematics.

A small Adjustment for factoring

A small adjustment is required to factor the difference of the two squares. It can be achieved by adding and subtracting a term $ab$ in the right-hand side of the algebraic equation.

$\implies$ $a^2-b^2$ $\,=\,$ $a^2-b^2-ab+ab$

$\implies$ $a^2-b^2$ $\,=\,$ $a^2-ab+ab-b^2$

Factorize the algebraic expression

$\implies$ $a^2-b^2$ $\,=\,$ $a^2-ab+ab-b^2$

The right-hand side of the equation can be factored by the factorisation method. $a$ is a common factor in the first two terms and $b$ is a common factor in the last two terms. So, they can be factored.

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

Now, $a-b$ is a common factor in the both terms of the right-hand side of the equation.

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

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

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