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) (Show Source): You can put this solution on YOUR website!
Suppose the roots are p and q. Therefore we want
 = (x-p)(x-q) = (x-p^2)(x-q^2))
.
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
 = x^2)
or  = x(x-1))
or  = (x-1)^2)
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
^2 = -\frac{1}{2} - \frac{i \sqrt{3}}{2})
(easy way to square it is note that p = e^(2i*pi/3))
Therefore the polynomial f(x) is
 = (x - (-\frac{1}{2} + \frac{i \sqrt{3}}{2}))(x - (-\frac{1}{2} - \frac{i \sqrt{3}}{2})) = x^2 - x - 1)
So there are four quadratic equations that satisfy (ignoring arbitrary constants we can multiply with).
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