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Nature of Roots

29.Show that ...

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29.Show that x+ax-4=0 has real and two distinct roots for all values of a.

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Q. Prove that the equation

x^{2} + p x - 1 = 0

has real and distinct roots for all real values of

p

| l n | x | | = | k - 1 | - 4

has exactly two distinct real roots for

x \in R - {0}

, then the sum of all possible integral values of k is

p (x) = x^{2} + a x + b

have two distinct real roots, where

a, b

are real numbers. Define

g (x) = p (x^{3})

for all real numbers

x

. Then which of the following statements are true?
I.

g

has exactly two distinct real roots
II.

g

can have more than two distinct real roots
III. There exists a real number

α

g (x) \geq α

x

.

P (x) = x^{2} + a x + 1

has two distinct real roots, then exhaustive values of

a

| l n | x | | = | k - 1 | - 4

has exactly two distinct real roots for

x \in R - {0}

, then the sum of all possible integral values of k is

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