Question

${(a + 2b)^2} - 8ab$ is equal to
(A) ${a^2} + 4{b^2}$ (B) ${a^2} - 4{b^2}$ (C) ${(a - 2b)^2}$ (D) ${a^2} + 2{b^2}$

Answer 1

Hint: To reduce the given expression we should try to apply some identity or should try to open the bracket. Here we will apply two basic square identities which are
${(a + b)^2} = {a^2} + {b^2} + 2ab$
${(a + b)^2} = {a^2} + {b^2} - 2ab$

Complete step by step answer:
By observing the given expression, we can conclude that an identity can be applied which is ${(x + y)^2} = {x^2} + {y^2} + 2xy$
On comparing the terms we get, $x = a$ and $y = 2b$, therefore
${(a + 2b)^2} - 8ab$
\[ = \{ {a^2} + {(2b)^2} + 2 \times a \times 2b\} - 8ab\] []
$ = {a^2} + 4b{}^2 + 4ab - 8ab$
$ = {a^2} + 4b{}^2 - 4ab$
\[ = {a^2} + {(2b)^2} - 2 \times a \times 2b\]
We know that,${m^2} + {n^2} - 2mn = {(m - n)^2}$ and if we assume that, $m = a$ and $n = 2b$, then
$ = {(a - 2b)^2}$
Hence, ${(a + 2b)^2} - 8ab = {(a - 2b)^2}$

Note:
  We have used square identities here, but in this type of questions to reduce the expressions there are more identities, for example:
${(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)$
${(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)$
We just have to find out which one is suitable to the given expression.