First slide

Theory of equations

Question

A quadratic equation whose roots are ${(\frac{γ}{α})}^{2}$ and ${(\frac{β}{α})}^{2}$ , where $α, β, γ$ are roots of $x^{3} + 27 = 0$ is

Moderate

A

$x^{2} - x + 1 = 0$
B

$x^{2} + 3 x + 9 = 0$
C

$x^{2} + x + 1 = 0$
D

$x^{2} - 3 x + 9 = 0$

Solution

Given that, $x^{3} + 27 = 0$
$\Rightarrow x^{3} = {(- 3)}^{3}$

$\Rightarrow x = - 3, - 3 ω, - 3 ω^{2}$

$∴$ $\frac{γ}{α} = \frac{- 3 ω^{2}}{- 3} \Rightarrow \frac{γ}{α} = ω^{2},$

And $\frac{β}{α} = \frac{- 3 ω}{- 3} \Rightarrow \frac{β}{α} = ω$

$∴$ ${(\frac{γ}{α})}^{2} = ω^{4} = ω$ and ${(\frac{β}{α})}^{2} = ω^{2}$

sum of the roots : ${(\frac{γ}{α})}^{2}$ + ${(\frac{β}{α})}^{2}$ $= ω + ω^{2}$

product of the roots : ${(\frac{γ}{α})}^{2}$ ${(\frac{β}{α})}^{2}$ $= ω^{3}$

$∴$ The required equation is $x^{2} - ({(\frac{γ}{α})}^{2} + {(\frac{β}{α})}^{2}) x + {(\frac{γ}{α})}^{2} {(\frac{β}{α})}^{2} = 0$

$\Rightarrow x^{2} - (ω + ω^{2}) x + ω^{3} = 0$

$\Rightarrow x^{2} + x + 1 = 0$ .

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