The quadratic equation ${x}^{2}-9x+3=0$ has roots $\alpha$ and $\beta$. If ${x}^{2}-bx-c=0$ has roots ${\alpha}^{2}$ and ${\beta}^{2}$, then $(b,c)$ is

Using the sum and product of roots formulas, alpha + beta = 9 and alpha*beta = 3. The new equation has roots alpha^2 and beta^2, so b equals their sum, which is (alpha+beta)^2 - 2*alpha*beta = 9^2 - 2(3) = 75. The product is c = (alpha*beta)^2 = 3^2 = 9, but the new equation is x^2 - bx - c = 0, so c is -9. This gives the pair (75, -9).

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