Let f(x) = x^2 - px + q = 0, and let α and β be the roots of f(x). Now, let y1 = α/β^2 and y2 = β/α^2 be the roots of a new equation g(y) = ay^2 + by + c = 0. We want to find the coefficients a, b, and c.
To find a, note that a * y1y2 = α/β^2 * β/α^2 = 1. Therefore, a = 1.
To find b, note that b * (y1+y2) = b * (α/β^2 + β/α^2) = α/β^2 * (α^2 + β^2). Since α and β are the roots of the given equation, we have the following relations:
1) α + β = p
2) αβ = q
We will use these relations to express (α^2 + β^2) in terms of p and q. Note that (α + β)^2 = α^2 + 2αβ + β^2 = (α^2 + β^2) + 2q, so (α^2 + β^2) = (α + β)^2 - 2q = p^2 - 2q. Therefore, b * (α/β^2 + β/α^2) = α/β^2 * (p^2 - 2q), which means that b = α/β^2 (p^2 - 2q).
To find c, note that c = - y1y2. Since y1 = α/β^2 and y2 = β/α^2, we have c = -α/β^2 * β/α^2 = -1.
So, the equation with roots α/β^2 and β/α^2 is g(y) = y^2 - α/β^2 (p^2 - 2q)y - 1 = 0.
If α and β are the roots of the equation x^2-px+q=0, form the equation whose roots are α/β^2 and β/α^2
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