# $(x+a)(x+b)$ identity

## Formula

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

$x$ is a variable, and $a$ and $b$ are constants. The literal $x$ formed a binomial $x+a$ with the constant $a$, and also formed another binomial $x+b$ with another constant $b$. The first term of the both sum basis binomials is same and it is a special case in algebraic mathematics. Therefore, the product of special binomials is called as the special product of binomials or special binomial product.

$(x+a)(x+b)$

The expansion of this special binomial product is equal to an algebraic expression $x^2+(a+b)x+ab$. Therefore, the product of special binomials $x+a$ and $x+b$ can be expanded as $x^2+(a+b)x+ab$ in algebra.

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

#### Use

This algebraic rule is used as a formula in mathematics if binomials are formed by summation and one of two terms of both binomials is same.

##### Example

$(m+2)(m+6)$ $\,=\,$ $m^2+(2+6)m+(2 \times 6)$

$\implies$ $(m+2)(m+6)$ $\,=\,$ $m^2+8m+12$

In fact, the product of binomials $(m+2)(m+6)$ can be obtained by the multiplication of algebraic expressions but it takes more time. Therefore, this standard result is used to find the product of them quickly.

### Proof

The expansion of special product of binomials $x+a$ and $x+b$ can be proved in mathematics in two different methods.

##### Algebraic Method

Learn how to derive the expansion of $(x+a)(x+b)$ identity by the multiplication of algebraic expressions in algebra.

##### Geometrical Method

Learn how to derive the expansion of $(x+a)(x+b)$ formula geometrically by the areas of rectangle and square in geometry.

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