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Grade: 11


if every pair of the equation x^2+px+qr=0 , x^2+qx+rp=0 , x^2+rx+pq=0 have a non zero common root then the sum of three common is ?

4 years ago

Answers : (3)

Sourabh Singh
IIT Patna
askIITians Faculty
2104 Points
4 years ago
Vikas TU
12276 Points
let a be the common root then satifying the root in all the three eqns. we get,
a^2 + ap + qr = 0
a^2 + aq + rp = 0
a^2 + ar + pq = 0
solving all three one by one by elimination method,
we get
a = r with p = q
a = p with q = r
a = q with p = r
Hence, sum of common root => p + q + r 
                                            => 3p = 3q = 3r respectively.
4 years ago
mycroft holmes
272 Points
Sourabh’s answer can be taken a bit further. Note that we have concluded that r is a root of the 1st quadratic i.e. x2+px+qr = 0. From product of roots, the other root is q. Sum of roots gives q+r = -p, or p+q+r = 0.
So, the sum of the common roots is 0
4 years ago
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