a =?

Algebra Level 3

If the roots of the equation x 3 6 a x 2 + 5 a x + 88 = 0 x^3 - 6ax^2+ 5ax + 88 = 0 forms an arithmetic sequence. If a a is a real constant, then what is the value of a = ? a=?

The answer is 2.

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2 solutions

Aryan Sanghi
Feb 1, 2018

Let the roots be z-d, z, z+d

Sum of roots = ( c o e f f i c i e n t o f x ² ) c o e f f i c i e n t o f x ³ \frac{-(coefficient of x²)}{coefficient of x³}



Putting x=2a.

(2a)³-6a(2a)²+5a(2a)+88=0 (Since z is a root of the equation and z=2a)





Thank you for sharing your solution.

Hana Wehbi - 3 years, 4 months ago
Srinivasa Gopal
Dec 24, 2017

As the three roots of the cubic equation are in a sequence the roots of the cubic equation

can be considered as r-d, r, r+d

So sum of roots = 3r = 6a

So r = 2a

The sum of roots taken two at a time = 5a can be written as

r (r-d) + r (r+d) + (r+d)(r-d) = r (2r) + r r - d d = 3 r r -d d

3 r r -d*d = 5a

So d d = 12 a a - 5 a

The product of the three roots is

(r r - d d)*r = -88

Rearranging all the three equations we get

   a*a(8a-5) = 44 = 4 * 11 and so a = 2;

Thank you for sharing your solution.

Hana Wehbi - 3 years, 5 months ago

Sir, there was no need of taking sum of roots taking 2 at a time, please see my solution :-)

Aryan Sanghi - 3 years, 4 months ago

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