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Find all the values of a for which the equations \(\begin{align*} x^2 + ax + 1 &= 0, \\ x^2  - x - a &= 0 \end{align*}\)have a common real root. Enter all the possible values, separated by commas.

 Jun 20, 2019
 #1
avatar+106519 
+1

x^2 + ax + 1  = x^2 - x - a

 

ax + 1  = -x - a

 

ax + x  =   -1 - a

 

x(1 + a)  =  - ( 1 + a)

 

x =   - (1 + a)

       _______

           (1 + a)

 

x = - 1    is the common root        [if a not equal to -1 ]

 

Therefore  using the first function

(-1)^2 + a(-1) + 1  = 0                     

1 - a + 1  = 0

2 - a  = 0

a = 2

 

And using the second function as a check

(-1)^2  - (-1) - 2  = 0     ???

1 + 1 - 2  = 0

2 - 2  = 0   true

 

 

cool cool cool

 Jun 20, 2019
 #2
avatar+1015 
+2

Nice one! 

Nickolas  Jun 20, 2019

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