• Algebra
• Trigonometry
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Proof of $(x+a)(x+b)$ formula in Algebraic Method

Formula

${(x+a)}{(x+b)}$ $\,=\,$ $x^2+(a+b)x+ab$

Proof

The expansion of special product of binomials $x+a$ and $x+b$ can be derived in algebraic method by the multiplication of algebraic expressions.

Multiplication of Algebraic Expressions

Multiply each term of first binomial with the second binomial to perform multiplication of algebraic expressions.

${(x+a)}{(x+b)}$ $\,=\,$ $x \times (x+b)$ $+$ $a \times (x+b)$

Now, multiply each term of the second binomial by its multiplying factor.

$=\,$ $x \times x$ $+$ $x \times b$ $+$ $a \times x$ $+$ $a \times b$

Arrange the product of terms in an order

Now, write product of the terms in an order to obtain the special product of binomials $x+a$ and $x+b$.

$=\,$ $x^2$ $+$ $xb$ $+$ $ax$ $+$ $ab$

$=\,$ $x^2$ $+$ $bx$ $+$ $ax$ $+$ $ab$

$=\,$ $x^2$ $+$ $ax$ $+$ $bx$ $+$ $ab$

Simplify the expansion of special product

$x$ is a common multiplying factor in two terms of the expression. So, take it common from them to express the expansion of $(x+a)(x+b)$ formula in algebraically.

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