lf $\alpha,\ \beta$ are the roots of the equation $\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}=0$, then the quadratic equation whose roots are $\alpha+\beta,\ \alpha\beta$ is

For the equation ax^2 + bx + c = 0, Vieta's formulas give the sum of the roots alpha + beta = -b/a and the product alpha*beta = c/a. The required equation must have roots equal to the sum and product of the original roots, so its sum of roots is (-b/a) + (c/a) = (c - b)/a and its product is (-b/a)(c/a) = -bc/a^2. Using the standard quadratic form x^2 - (sum)x + (product) = 0, we get x^2 - ((c - b)/a)x - bc/a^2 = 0. Multiplying the entire equation by a^2 to clear the denominators yields a^2x^2 + a(b - c)x - bc = 0.

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