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Nature of Roots

If x^2-2px+q=...

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If x^2-2px+q=0 has two equal roots then the equation (1+y)x^2 -2(p+y)x +(q+y)=0 will have real and distinct roots when

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Q. If the equation

x^{2} - 2 p x + q = 0

has two equal roots, then the equation

(1 + y) x^{2} - 2 (p + y) x + (q + y) = 0

will have its roots real and distinct only when

m, n

be positive integers and the quadratic equation

4 x^{2} + m x + n = 0

has two distinct real roots

p

q

(p \leq q)

. Also, the quadratic equations

x^{2} - p x + 2 q = 0

x^{2} - q x + 2 p = 0

have a common root say

α

p

q

are rational, then uncommon root of the equation

x^{2} - p x + 2 q = 0

x^{2} - q x + 2 p = 0

Q. If p, q, r are real, then roots of the equation

x^{2} - 2 p x + p^{2} - q^{2} - r^{2} = 0

will always be?

tan 25^{\circ}

tan 20^{\circ}

are the roots of

x^{2} + 2 p x + q = 0

, then the value of

2 p - q

α

β

x^{2} + p x + q = 0

α^{4}, β^{4}

x^{2} - r x + s = 0

, then prove that the equation

x^{2} - 4 q x + 2 q^{2} - r = 0

has distinct and real roots.

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