If α and β are the roots of the quadratic equation ax2 + bx + c = 0,

1.	If α and β are the roots of the quadratic equation ax2 + bx + c = 0, such that β = α1/3, then(a) (a3b)1/4 + (ac3)1/4 + a = 0(b) (a3c)1/4 + (ac3)1/4 + b = 0(c) (a3b)1/4 + (ab3)1/4 + c = 0(d) (a3c)1/4 + (bc3)1/4 + a = 0
Answer» (b) (ac³)^1/4 + (a³c)^1/4 + b = 0 Let α, β be the roots of the equation ax² + bx + c = 0. Then, α + β = \(-\frac{b}{a}\) .....(i) αβ = \(\frac{c}{a}\) ......(ii) and β = α^1/3 .......(iii) ∴ From (ii) and (iii), α. (α)^1/3= \(\frac{c}{a}\) ⇒ α^4/3= \(\frac{c}{a}\) ⇒ α = \(\bigg(\frac{c}{a}\bigg)^{3/4}\) ∴ β = \(\bigg(\big(\frac{c}{a}\big)^{3/4}\bigg)^{1/3}\) = \(\bigg(\frac{c}{a}\bigg)^{1/4}\) ∴ Putting these values of a and b in eqn. (i), we have \(\bigg(\frac{c}{a}\bigg)^{3/4}\) + (c/a)^1/4 = \(-\frac{b}{a}\) ⇒ a. a^–3/4 c^3/4 + a. a^–1/4 c^1/4 = – b ⇒ a^1/4 c^3/4 + a^3/4 c^1/4 + b = 0 ⇒ (ac³)^1/4 + (a³c)^1/4 + b = 0.

If α and β are the roots of the quadratic equation ax2 + bx + c = 0, such that β = α1/3, then(a) (a3b)1/4 + (ac3)1/4 + a = 0(b) (a3c)1/4 + (ac3)1/4 + b = 0(c) (a3b)1/4 + (ab3)1/4 + c = 0(d) (a3c)1/4 + (bc3)1/4 + a = 0

Answer»

(b) (ac³)^1/4 + (a³c)^1/4 + b = 0

Let α, β be the roots of the equation ax² + bx + c = 0. Then,

α + β = \(-\frac{b}{a}\) .....(i)

αβ = \(\frac{c}{a}\) ......(ii)

and β = α^1/3 .......(iii)

∴ From (ii) and (iii), α. (α)^1/3= \(\frac{c}{a}\) ⇒ α^4/3= \(\frac{c}{a}\) ⇒ α = \(\bigg(\frac{c}{a}\bigg)^{3/4}\)

∴ β = \(\bigg(\big(\frac{c}{a}\big)^{3/4}\bigg)^{1/3}\) = \(\bigg(\frac{c}{a}\bigg)^{1/4}\)

∴ Putting these values of a and b in eqn. (i), we have

\(\bigg(\frac{c}{a}\bigg)^{3/4}\) + (c/a)^1/4 = \(-\frac{b}{a}\)

⇒ a. a^–3/4 c^3/4 + a. a^–1/4 c^1/4 = – b

⇒ a^1/4 c^3/4 + a^3/4 c^1/4 + b = 0

⇒ (ac³)^1/4 + (a³c)^1/4 + b = 0.