If αand βare the roots of x^2+p x+q=0 and α^4 and β^4 are the roots of x^2-r x+s=0, then the equ

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If αand βare the roots of x^2+p x+q=0 and α^4 and β^4 are...

If $\alpha$ and $\beta$ are the roots of $x^{2}+p x+q=0$ and $\alpha^{4}$ and $\beta^{4}$ are the roots of $x^{2}-r x+s=0$, then the equation $x^{2}-4 q x+2 q^{2}-r=0$ has always
(A) two real roots
(B) two positive roots
(C) two negative roots
(D) one positive and one negative root

Key Concepts

Discriminant Analysis

Discriminant analysis examines the expression under the square root in the quadratic formula. It determines the nature (real and distinct, real and equal, or complex) of the roots. This analysis is crucial in understanding the sign and number of the roots of a quadratic equation by investigating the relationships among the coefficients.

Quadratic Equations

This concept deals with second?degree polynomials and their properties. It includes the structure of a quadratic equation, the conditions for the existence of real roots, and techniques for solving them. Understanding quadratic equations is essential because many problems in algebra involve determining the nature of roots from the coefficients of the equation.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. In the context of quadratic equations, these formulas provide a direct link between the roots and the coefficients, enabling the expression of symmetric functions of the roots in terms of the equation's coefficients. This is a powerful tool in problems where roots undergo transformations or when new equations are formed from functions of those roots.

Symmetric Functions and Power Sums

This concept involves expressing functions of the roots, such as the sum of their powers, as functions of the elementary symmetric sums given by the coefficients. It’s particularly useful when dealing with expressions like the fourth powers of roots. Understanding symmetric functions helps in reducing complex expressions into simpler forms using known relationships among the roots.

Transformation of Roots

This concept focuses on how operations performed on the roots of an equation, such as raising them to a power, result in new sets of roots that satisfy another polynomial equation. It is key when one needs to derive new relationships between the coefficients of the original equation and those of the new equation constructed from the transformed roots.

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