Latest TCS Aptitude Question SOLUTION: In a quadratic equation, (whose coefficients are not necessarily real) the constant term is not 0. The cube of the sum of the squares of its roots is equal to the square of the sum

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In a quadratic equation, (whose coefficients are not necessarily real) the constant term is not 0. The cube of the sum of the squares of its roots is equal to the square of the sum of the cubes of its roots. Which of the following is true?
option
a) Both roots are real
b) Neither of the roots is real
c) At least one root is non-real
d) At least one root is real

Asked In TCS Tanu (11 years ago)

Unsolved Read Solution (15)

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Read Solution (15) :

Ashutosh Anshu  (11 years ago)

let p and q are the roots. thus a/c

(p^2+q^2)^3 = (p^3+q^3)^2
solving both we get, p^2*q^2(3p^2+3q^2-2pq) = 0
now 3p^2+3q^2-2pq = 0.
p^2-(2/3*p*q)+q^2 = 0. thus here p and q are not real.

Is this solution Helpfull? Yes (10) | No (3)  

AMIT KUMAR  (11 years ago)

(b)is the correct option.bcz after calculating the values of the root we don't get any real roots.

Is this solution Helpfull? Yes (5) | No   

jit dhar  (11 years ago)

i think 'a' is correct isn't it tanu.

Is this solution Helpfull? Yes (2) | No (4)  

Apoorv Shrivastava  (10 years ago)

Assume the given quadratic equation is ax2+bx+c=0 whose roots are p, q.
Now given that (α2+β2)3=(α3+β3)2
By expanding we get, α6+3.α4.β2+3.α2.β4+β6=α6+β6+2.α3.β3
3.α2.β2(α2+β2)=2.α3.β3
3.(α2+β2)=2.α.β
3.(α2+β2)+6.α.β−6.α.β=2.α.β
3.(α+β)2=8.α.β ...(1)
We know that sum of the roots = α+β=−ba
product of the roots = α.β=ca
Substituting in the equation (1) we get 3.(−ba)2=8.ca⇒3.b2=8.a.c

The nature of the roots can be determined by finding the magnitude of the determinant = b2−4ac
But we know that ac = 3b28
So b2−4ac = b2−4.3b28=−b88

Is this solution Helpfull? Yes (2) | No   

AMIT KUMAR  (11 years ago)

@Tanu actually i hav solved and it is very large so i can't elobarte it. but finally i got the conclusion is that neither of them are real. and most important i want to know that how can u say that my ans is wrong.

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Ashutosh Anshu  (11 years ago)

@Tanu singh - Then solve it and Explain it.

Is this solution Helpfull? Yes (1) | No (1)  

ayushi garg  (11 years ago)

@tanu..can u tell me the correct answer of it n explain its solution too..??

Is this solution Helpfull? Yes (1) | No   

Akhilesh Agrawal  (11 years ago)

tanu if you know answer at-least say what is ans

Is this solution Helpfull? Yes (1) | No   

Ankit Verma  (11 years ago)

Yes Tanu say atleast what is ans.

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Tanu  (11 years ago)

@AMIT The answer you gave is wrong but still please explain how reached on your conclusion?

Is this solution Helpfull? Yes | No (1)  

Tanu  (11 years ago)

@Amit Acually i know the correct option. I have answers ..i just need explanation.

Is this solution Helpfull? Yes | No (1)  

Tanu  (11 years ago)

@Ashutosh

As per your solution option b suits but its not correct.

Is this solution Helpfull? Yes | No (3)  

Tanu  (11 years ago)

@Everyone (c) is the correct option.

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Ajay  (9 years ago)

Roots are imaginary

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Praful amzare  (8 years ago)

Both root are imaginary so option 2 is correct

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