In the equation x^2 + bx + c = 0, where b and c are constants, what is

Concepts tested here are:

For a Quadratic equation \(x^2 + bx + c = 0\)
1. Sum of roots = \(\frac{-b}{a}\\
\)
2. Product of roots = \(\frac{c}{a}\)

We need to find the value of "b".

Statement 1: Product of the roots = 1 = \(\frac{c}{a}\). Since, a = 1 for the given equation. We get c = 1. ----------(1)
Insufficient to find the value of 'b'.

Statement 2: Sum of the cubes of roots = \(\frac{65}{8}\)

We know sum of roots = -b/a = -b (since a is equal to 1 in this case)

Let the roots be r1 and r2.
So, r1 + r2 = -b.
Take cube on both side,
You get \((r1 + r2)^3 = -b^3\)
=> \(r1^3 + r2^3 + 3(r1 + r2)(r1*r2) = -b^3\)
We know from statement 2 the value of \(r1^3 + r2^3\) =\(\frac{ 65}{8}\). Also, (r1 + r2) is nothing but sum of roots = \(\frac{-b}{a}\). But, we do not know the value of r1*r2, so statement 2 alone is insufficient. But, we know the value of r1*r2 from statement 1.

Substitute these values in the equation and you will see that only 'b' is the unknown in the equation. Upon solving which you will get the answer.

Hence both statements together are sufficient.