The equation, whose roots are the squares of the roots of the equation ax2 – bx + c = 0, is
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a2x2 + b2x + c2 = 0
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a2x2 – (b2 – 4ac)x + c2 = 0
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a2x2 – (b2 – 2ac)x + c2 = 0
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a2x2 + (b2 – ac)x + c2 = 0
If roots of ax^2 - bx + c = 0 are alpha and beta, then alpha+beta = b/a and alpha*beta = c/a. The new equation has roots alpha^2 and beta^2. Sum = alpha^2 + beta^2 = (alpha+beta)^2 - 2*alpha*beta = (b/a)^2 - 2(c/a) = (b^2 - 2ac)/a^2. Product = (alpha*beta)^2 = c^2/a^2. Equation: x^2 - (Sum)x + Product = 0 => x^2 - ((b^2-2ac)/a^2)x + c^2/a^2 = 0 => a^2x^2 - (b^2-2ac)x + c^2 = 0.
Let the roots of the original equation be p and q, so by Vieta's formulas p + q = b/a and pq = c/a. The new equation has roots p^2 and q^2, so its sum of roots is p^2 + q^2 = (p + q)^2 - 2pq. Substituting the Vieta values gives the sum as (b/a)^2 - 2(c/a) = (b^2 - 2ac)/a^2, and the product of the new roots is (pq)^2 = c^2/a^2. Multiplying the variable x^2 by a^2 to clear the denominators results in the equation a^2x^2 - (b^2 - 2ac)x + c^2 = 0.