Question

(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are alawys real if they will be equal if a=b=c prove

Answer 1

Simplifing the equation we get
3x^-2x(a+b+c)+(ab+bc+ca)=0
,ATQ
D=0
I.e [-2(a+b+c)]^2-4x3(ab+bc+ca)=0
4(a+b+c)^2=4x3(ab+bc+ca)
(a+b+c)^2=3(ab+bc+ca)
a^2+b^2+c^2+2(ab+bc+ca)=3(ab+bc+ca)
a^2+b^2+c^2=ab+bc+ca ………(2)
And equation (2) is only possible when a=b=c
Now we will assume
That a=b=c=k
Then if we replace every value in equation (2)
L.H.S
K*k+k*k+k*k
R.H.S
K*k+k*k+k*k
R.h.s=L.h.s
Hence proved
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