Alzebra : 1021

Problem 1

The equations x² + ax + 1 = 0 and x² + bx + c = 0 have a common real root, and the equations x² + x + a = 0 and x² + cx + b = 0 have a common real root. Find a + b + c.

 

Solution

The common root of x² + ax + 1 = 0 and x² + bx + c = 0 must also satisfy (a-b)x + (1-c) = 0, 

so it must be (c-1)/(a-b). 

Note that the other root of x² + ax + 1 = 0 must be (a-b)/(c-1), since the product of the roots is 1. 

Similarly the common root of x² + x + a = 0 and x² + cx + b = 0 must satisfy (c-1)x + (b-a) = 0, 

so it is x = (a-b)/(c-1). 

Hence x² + x + a = 0 and x² + ax + 1 = 0 have a common root. 

Hence it satisfies (a-1)x + (1-a) = 0. Now we cannot have a = 1, for then x² + ax + 1 has no real roots. Hence the common root must be 1. 

Hence both roots of x² + ax + 1 = 0 are 1 and so a = -2.

So x² + bx + c = 0 has one root 1. 

Then its other root must be c/1 = c. Hence -b = 1 + c, or b + c = -1. Hence a + b + c = -3. 

Answer

-3