# 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.

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

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