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Nature of Roots

Prove that ro...

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Prove that roots of the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are always real and can be equal unless a=b=c

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Q. If a, b and c are real, then both the roots of the equation

(x - b) (x - c) + (x - c) (x - a) + (x - a) (x - b) = 0

Q. Show that the roots of the equations

(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0

and they cannot be equal unless

a = b = c

Q. Both roots of the equation (x - a) (x - b) + (x - b) (x - c) + (x - c)(x - a) = 0 are always

Q. Both the roots of the given equation

(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0

Q. (a) Prove that the roots of

(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0

are always real and they will be equal if and only if

a = b = c

.
(b) Examine the nature of the roots of the quadratic

{(b - x)}^{2} - 4 (a - x) (c - x) = 0

where a,b,c are real.
(c) Discuss the nature of the roots of the equation

x^{2} + 2 (3 λ + 5) x + 2 (9 λ^{2} + 25) = 0

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