Question

What should be added to $ {x^2} + \dfrac{1}{{25}}{x^2}{y^2} $ to make it a perfect square?

Answer 1

Hint: Here in this question we have to find what we have to add such that it can form a perfect square. So we will try to make in the form of $ {(a + b)^2} $ for that we have to find what should be added such that it can be written as $ {(x + \dfrac{1}{5}xy)^2} $ .

Complete step-by-step answer:
Given $ {x^2} + \dfrac{1}{{25}}{x^2}{y^2} $
We have to find what should be added in the above expression such that it can make a perfect square.
We have $ {a^2} + {b^2} $ we need $ 2ab $ to make it perfect square
Here $ {x^2} = {a^2} $ and $ \dfrac{1}{{25}}{x^2}{y^2} = {b^2} $
So we can now say that we have to add a 2ab term in the given expression.
Here $ a = x $ and $ b = \dfrac{1}{5}xy $
So the \[\]
  $
\Rightarrow 2ab = 2 \times x \times \dfrac{1}{5}xy \\
\Rightarrow 2ab = \dfrac{2}{5}{x^2}y \\
 $
So we have to add $ \dfrac{2}{5}{x^2}y $
Such that we can write in the perfect square form
 $ {\left( {x - \dfrac{1}{5}xy} \right)^2} $
Similarly we can write it as $ {\left( {x - \dfrac{1}{5}xy} \right)^2} $
Here we have subtracted $ \dfrac{2}{5}{x^2}y $
Hence we have to add $ \dfrac{2}{5}{x^2}y $ to make the given expression a perfect square.

Note: Students please be aware do not consider $ \dfrac{2}{5}{x^2}y $ as 2ab otherwise you will get confused and you will get the wrong answer as well.