Roots You Never Solve For

Question

Let r and s be the roots of x^2 - 7x + 12 = 0. Without computing r or s individually, evaluate frac1r + frac1s exactly. Your method should survive unchanged if the constant 12 is replaced by 11 and the roots turn irrational.

Accepted Answer

Vieta's formulas. Expanding (x - r)(x - s) = x^2 - (r+s)x + rs and matching coefficients with x^2 - 7x + 12: r + s = 7, qquad rs = 12. Combine. The target is symmetric in the roots, so it must be expressible through those two numbers: frac1r + frac1s = fracr + srs = frac712 approx mathbf0.583. Check (allowed after the fact): the quadratic factors as (x-3)(x-4), and frac13 + frac14 = frac712. Agreement. Why the method is the point. Swap 12 for 11 and the roots become frac7 pm sqrt52, ugly, but the answer is instantly frac711: the symmetric-function route never touched the roots. The underlying fact (fundamental theorem of symmetric polynomials) is that every symmetric expression of the roots is a polynomial in the elementary symmetric functions e_1 = r+s and e_2 = rs. For example r^2 + s^2 = e_1^2 - 2e_2 = 49 - 24 = 25 here. A slick companion fact: the polynomial whose roots are frac1r and frac1s is the original with coefficients reversed, 12x^2 - 7x + 1 = 0, and its "sum of roots" coefficient ratio frac712 is the same answer read directly. The desk relevance is real: linear recursions and AR(2) processes are governed by the roots of a characteristic quadratic, and stability or stat

Roots You Never Solve For

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