Math Doubts

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