find the value of a

Question

x²+ax+1=0 and x²+x+a=0 have roots. What is the value of a?

 

 

 

Answer 1

Rather than factor, first subtract equations:

   x² + ax + 1 = 0

   x² +  x  + a = 0

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       (a-1)x +(1-a) = 0

 

       (a-1)x - (a-1) = 0

       (a-1)x = (a-1)

          x = 1

 

So,

  1 + a + 1 = 0

 and

  1 + 1 + a = 0

 

That means:   a = -2   and x must be 1.

 

   x² -2x + 1 = 0

    (x-1)(x-1) = 0

      has roots x=1 and x=1

 

   x² + x -2 = 0

   (x+2)(x-1) = 0

      has roots x=1 and x=-2

 

For both equations to be true simultaneously, x=1 and a=-2.