SOLUTION: if a^2 + 1 = a, then the value of a^12 + a^6 +1 is a)2 b)3 c)-3 d)1

Question 1113784: if a^2 + 1 = a, then the value of a^12 + a^6 +1 is
a)2
b)3
c)-3
d)1

Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.

If  a^2 + 1 = a,  then  a^2 - a + 1 = 0.


    Multiply both sides by (a+1). You will get   a^3 + 1 = 0,   or   a^3 = -1.


    So, the values of "a" that are the roots of the original equation, are the complex cubic roots of (-1).


          // If you solve the original equation using the quadratic formula, you will get the same result.



    Therefore,  a^12 = (a^3)^4 = (-1)^4 = 1,   and

                a^6  = (a^3)^2 = (-1)^2 = 1.


    Thus  a^12 + a^6 + 1 = 1 + 1 + 1 = 3.


The answer is 3.  Choice b).

Solved.

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