$(a+b)^2 = a^2+b^2+2ab$
$a$ 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.
$(1) \,\,\,$ 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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