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Nature of Roots

if alpha and ...

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if alpha and beta are the roots of the quadratic equation 3 x square - 4 x + 1 is equal to zero then find the quadratic equation whose roots are 1) alpha/beta and beta/alpha 2) (alpha)^2/ beta and (beta)^2/ alpha

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α

β

are roots of the quadratic equation

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

then quadratic equation whose roots are

\frac{α - 1}{α + 1}, \frac{β - 1}{β + 1}

α

β

are roots of quadratic equation

x^{2} - 1 = 0

, then find the equation whose roots are

\frac{2 α}{β}

\frac{2 β}{α}

α + β = 5

α^{3} + β^{3} = 35

, find the quadratic equation whose roots are

α

β

α, β

are the roots of

x^{2} + 3 x + 4 = 0,

then the quadratic equation whose roots are

α^{3} + 4 α^{2} + 7 α + 7

β^{3} - β^{2} - 8 β - 14

α, β

are two real and different roots of a quadratic equation and the determinant

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 4 - α & 3 & 4 - β α & 1 & β α^{3} & 1 & β^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

(α, β \neq 1) .

Then the quadratic equation can be

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