Correct Answer - Option 3 : Imaginary.
Concept:
The solution to the quadratic equation Ax2 + Bx + C = 0 can also be given by: \(\rm x=\frac{-B\pm \sqrt{B^2-4AC}}{2A}\).
The quantity B2 - 4AC is also called the discriminant.
- If B2 - 4AC ≥ 0, the roots are real.
- If B2 - 4AC = 0, the roots are real and equal.
- If B2 - 4AC < 0, the roots will be complex and conjugates of each other.
The sum of both the roots of the quadratic equation Ax2 + Bx + C = 0 is \(\rm -\frac{B}{A}\) and the product of the roots is \(\rm \frac{C}{A}\).
Calculation:
Using the expressions for the sum and the product of the roots, we have:
α + β = -p ... (1)
αβ = \(\rm \frac{p^2}{2}\) ... (2)
α4 + β4 = r ... (3)
Squaring equation (1), we get:
α2 + β2 + 2αβ = p2
Using equation (2), we get:
⇒ α2 + β2 = 0
Squaring again, we get:
⇒ α4 + β4 + 2α2β2 = 0
Using equations (2) and (3), we get:
⇒ r = -\(\rm \frac{p^4}{2}\) ... (4)
The discriminant of the equation 2x2 - 4p2x + 4p4 - 2r = 0 is:
(-4p2)2 - 4(2)(4p4 - 2r)
= 16p4 - 32p4 + 16r
Using equation (4), we get:
= 16p4 - 32p4 - 8p4
= -24p4, which is always negative for non-zero real p.
Since the discriminant is < 0, the roots are imaginary.