Math Doubts

$(a+b)^2$ formula

Formula

$(a+b)^2 = a^2+b^2+2ab$

Introduction

$a$ and $b$ are two variables but represent two terms. The sum of them is equal to $a+b$, which is a binomial. The square of the binomial $a+b$ is written as $(a+b)^2$, which is also known as the square of sum of two terms.

The $a+b$ whole square is used as a formula to expand it as an algebraic expression $a^2+2ab+b^2$ in mathematics.

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

Usage

The square of sum of two terms formula is used in two different cases in mathematics.

  1. It is used to expand the square of sum of two terms or a binomial.
  2. If any mathematical expression is in the form of $a^2+b^2+2ab$, then it is simply written as $(a+b)^2$.

Examples

$(1) \,\,\,$ Find $(3x+4y)^2$

Now, take $a = 3x$ and $b = 4y$ and substitute them in the expansion of the formula for evaluating its value.

$\implies$ $(3x+4y)^2$ $\,=\,$ $(3x)^2+(4y)^2+2(3x)(4y)$

$\implies$ $(3x+4y)^2$ $\,=\,$ $9x^2+16y^2+2 \times 3x \times 4y$

$\implies$ $(3x+4y)^2$ $\,=\,$ $9x^2+16y^2+24xy$

$(2) \,\,\,$ Simplify $p^2+25q^2+10pq$

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $p^2+(5q)^2+10pq$

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $p^2+(5q)^2+2 \times 5 \times p \times q$

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $p^2+(5q)^2+2 \times p \times 5 \times q$

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $p^2+(5q)^2+2 \times p \times 5q$

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $p^2+(5q)^2+2(p)(5q)$

Now, take $a = p$ and $b = 5q$, and simplify the algebraic expression by the $(a+b)^2$ identity

$\implies$ $p^2+25q^2+10pq$ $\,=\,$ $(p+5q)^2$

Proofs

The $a$ plus $b$ whole square identity can be derived in mathematics in two different methods.

  1. Learn the algebraic approach to derive the expansion of the $a+b$ whole square formula by the multiplication.

  2. Learn the geometric method to derive the expansion of the $a+b$ whole squared identity by the areas of geometric shapes.

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