Quantitative Aptitude Ques 1920

Question: What should be added to the $ x(x+a)(x+2a) $ $ (x+3a), $ so that the sum be a perfect square?

Options:

A) $ a^{2} $

B) $ a^{4} $

C) $ a^{3} $

D) $ a^{6} $

Show Answer

Answer:

Correct Answer: B

Solution:

  • $ x(x+a)(x+2a)(x+3a) $ $ =(x^{2}+ax)(x^{2}+5ax+6a^{2}) $ $ =x^{4}+ax^{3}+5ax^{3}+5a^{2}x^{2}+6a^{2}x^{2}+6a^{3}x $ $ =x^{4}+ax(x^{2}+5x^{2}+5ax+6ax+6a^{2}) $ $ =x^{4}+ax(6x^{2}+11ax+6a^{2}) $ … (i) For terms to be perfect square, $ {{(x+y)}^{2}}{{(x+y)}^{2}} $ $ =(x^{2}+2xy+y^{2})(x^{2}+y^{2}+2xy) $ $ =x^{4}+2x^{3}y+x^{2}y^{2}+x^{2}y^{2}+2xy^{3}+y^{4} $ $ +2x^{3}y+4x^{2}y^{2}+2xy^{3} $ $ =x^{4}+xy(4x^{2}+6xy+4y^{2})+y^{4} $ … (ii) On comparing Eqs. (i) and (ii), $ y=a $ So, $ a^{4} $ must be added to make it a perfect square.
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