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Roots of a Quadratic Equation

If alpha, bet...

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If alpha, beta, gamma, delta are the roots of the equation x^4 + ax^3 + bx^2 + cx + d = 0, find the value of summation of alpha^2*beta

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α, β, γ

be the roots of

a x^{3} + b x^{2} + c x + d = 0

, then find the value of :

\sum α^{2}

\sum \frac{1}{α}

\sum α^{2} (β + γ)

α, β, γ

are the roots of

a x^{3} + b x^{2} + c x + d = 0

∣ ∣ ∣ ∣ ∣ \begin{matrix} α & β & γ β & γ & α γ & α & β \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

α \neq β \neq γ

, then find the equation whose roots are

α + β - γ, γ + α - β, β + γ - α

α, β, γ

are the roots of the cubic polynomial equation

x^{3} + b x^{2} + c x + d = 0,

α β + β γ + γ α = - c

α, β, γ

Δ

are the roots of the equation

x^{4} - 1 = 0

, then the value of

\frac{a α + b β + c γ + d Δ}{a γ + b Δ + c α + d β} + \frac{a γ + b Δ + c α + d β}{a α + b β + c γ + d Δ}

Q. Assume the biquadratic

x^{4} - a x^{3} + b x^{2} - a x + d = 0

has four real roots with

\frac{1}{2} < α, β, γ, δ \leq 2.

Maximum possible value of

\frac{(α + β) (α + γ) δ}{(δ + β) (δ + γ) α}

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Roots of a Quadratic Equation

Standard XII Mathematics

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