If 6 + i and a^2 - a + b*i are the roots of the same quadratic equation, what is the value of a and b?

If a quadratic equation ax^2 + bx + c = 0, has one complex
root a + bi, the other root has to be equal to a – bi.

Here
one of the roots is 6 + i, so the other root is 6 – i.

Now
6 – i = a^2 - a + b*i

Equating the real
coefficient

a^2 – a =
6

=> a^2 – a – 6 =
0

=> a^2 – 3a + 2a – 6
=0

=> a (a – 3) + 2(a – 3) =
0

=> (a + 2) (a – 3) =
0

So a can be -2 and 3.

Also,
equating the imaginary coefficient b =
-1

Therefore a is equal to -2 and 3 and b is
-1.