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

$a$ 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.

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

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

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

$(1) \,\,\,$ 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

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

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

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

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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