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Nature of Roots

if the roots ...

Question

if the roots of quadratic equation a[b-c]x^2+b[c-a]x+c[a-b]=0 are equal and a,b,c>0 then prove that a,b,c are in hp

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Q. If the roots of the quadratic equation (a-b)x²+(b-c)x+(c-a)=0 are equal prove that 2a=b+c.

Q. If roots of quadratic equation

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

are real and equal then prove that

2 b = a + c

Q. (a) If the roots of the equation,

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

be equal, then prove thar a,b,c are in arithmetical progression.
(b) If

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

has equal roots, prove that a,b,c are in harmonical progression.

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