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Nature of Roots

If alpha, bet...

Question

If alpha, beta are roots of ax2 + bx + c = 0, then the equation ax2 - bx (x - 1) + c(x - 1)2 = 0 has roots 1) alpha/(1-alpha), beta/(1-beta) 2) (1- alpha)/alpha, (1-beta)/beta 3) alpha/(1+alpha), beta/(1+beta) 4) (1+alpha)/alpha, (1+beta)/beta

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α, β

are the roots of

a x^{2} + b x + c = 0

, then find the equation whose roots are

\frac{1}{a^{2}}, \frac{1}{β^{2}}

\frac{1}{a α + β}, \frac{1}{a β + b}

α + \frac{1}{β}, β + \frac{1}{α}

Q. If roots of the equation

a x^{2} + b x + c = 0

α, β

, then the equation whose roots are

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

α \neq 1, β \neq 1

a x^{2} + b x + c = 0

α

β

a α^{2} + b α + c = 0

a β^{2} + b β + c = 0

α, β

are the roots of

x^{2} - p (x + 1) - c = 0, c \neq 1

\frac{α^{2} + 2 α + 1}{α^{2} + 2 α + c} + \frac{β^{2} + 2 β + 1}{β 2 + 2 β + c} =

α

β

roots of the equation

a x^{2} + b x + c = 0

(1 + α + α^{2}) (1 + β + β^{2})

α

β

are the roots of the equation

a x^{2} + b x + c = 0; a, b, c \in I^{+}

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 1 + α + β & 1 + α^{2} + β^{2} 1 + α + β & 1 + α^{2} + β^{2} & 1 + α^{3} + β^{3} 1 + α^{2} + β^{2} & 1 + α^{3} + β^{3} & 1 + α^{4} + β^{4} \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ

is always divisible by

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