SOLUTION: how many quadratic equations are there whose roots when squared gives the same quadratic equation

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Question 577452: how many quadratic equations are there whose roots when squared gives the same quadratic equation
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose the roots are p and q. Therefore we want

.

Hence either p = p^2 and q = q^2, or p = q^2 and q = p^2. The first case implies that p and q must be either 0 or 1, so we have

or or

The second case implies p = p^4, or p^3 = 1. The roots for p are 1, -1/2 + i*sqrt(3) and -1/2 - i*sqrt(3)/2 (same for q).

We have taken care of the p = 1 case, so suppose that p is -1/2 + i*sqrt(3)/2. Then

(easy way to square it is note that p = e^(2i*pi/3))

Therefore the polynomial f(x) is



So there are four quadratic equations that satisfy (ignoring arbitrary constants we can multiply with).