# What is to be added to (x+a)(x+3a)(x+5a)(x+7a) so as to make it a perfect square

2
by anudeepd2

2014-08-11T21:27:11+05:30
As you can observe that
(x+a)(x+7a) =
and
(x+3a)(x+5a) =

If we can make the  terms equal, it will be perfect square.

(1) subtract

(1) Subtract

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Anudeep wait... i'll be commenting soon, actually yup, we should answer in constant terms not in term of "x"
Just multiply the all the factors and assume that it is of the form (x^2 + 8ax + ca^2)^2 for some constant c. Now compare both the expression and solve it for c as i did in last question you asked. this will give c = 11. Thus the constant term is c^2 * a^4 so it equal to 121a^4 and in the question we have 105a^4 so we need to add 16a^4
I hope you can do the comparisons as I did it in one of the question you asked
2014-08-11T22:46:42+05:30
(x+a)(x+7a)(x+3a)(x+5a)
(x²+8ax+7a²)(x²+8ax+15a²)
now if they are to be perfect square then we must make each expression a sqaure by themselves so we add and subtract by the opposite
(x²+8ax+7a²)(x²+8ax+15a²)-(x²+8ax+7a²)8a²
taking common we have
(x²+8ax+7a²)(x²+8ax+15a²-8a²)
(x²+8ax+7a²)(x²+8ax+7a²)= (x²+8ax+7a²)²
similarly we have
(x²+8ax+7a²)(x²_8ax+15a²)+(x²+8ax+15a²)8a²
(x²+8ax+15a²)(x²+8ax+7a²+8a²)= (x²+8ax+15a²)²