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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 ?
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 Sourabh Singh
IIT Patna
2104 Points
``` ```
4 years ago
```							let a be the common root then satifying the root in all the three eqns. we get,a^2 + ap + qr = 0a^2 + aq + rp = 0a^2 + ar + pq = 0solving all three one by one by elimination method,we geta = r with p = qa = p with q = ra = q with p = rHence, sum of common root => p + q + r                                             => 3p = 3q = 3r respectively.
```
4 years ago
```							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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