The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has

Engineering

Mathematics

Quadratic Equation and Nature of Roots

Question

neither real nor purely imaginary roots

two real and two purely imaginary roots

all real roots

only purely imaginary roots

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Let p(x) = Ax² + Bx + C

If p(x) = 0 has purely imaginary roots ⇒ sum of roots = 0

$∴$ B = 0

Also, D < 0

⇒ – 4AC < 0 ⇒ AC > 0

p(x) = Ax² + C

Now, p(p(x)) = A (p(x))² + C = A (Ax² + C)² = A (A²x⁴ + 2ACx² + C²) + C

$∴$ p(p(x)) $\equiv$ A³x⁴ + 2A²Cx² + (AC² + C) = 0

$\Rightarrow x^{2} = \frac{- 2 A^{2} C \pm \sqrt{4 A^{4} C^{2} - 4 A^{3} ({AC}^{2} + C)}}{2 A^{3}}$

$x^{2} = (\frac{- 2 A^{2} C \pm \sqrt{- 4 A^{3} C}}{2 A^{3}})$

As, AC > 0

⇒ roots of p(p(x)) = 0 are neither real nor purely imaginary roots.