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Question

The quadratic x^2 + 32 x - 1 has the following property: the roots, which are 12 and -2, are one less than the final two coefficients. Find a quadratic with leading term x^2 such that the final two coefficients are both non-zero, and the roots are one more than these coefficients. Enter the roots of this quadratic.

Accepted Answer

Let the quadratic be x^2 + ax + b.Then the roots are a + 1 and b + 1. By Vieta's formulas, align* (a + 1) + (b + 1) &= -a, (a + 1)(b + 1) &= b. align*From the first equation, a + 1 = -b2. Substituting into the second equation, we get -b2 (b + 1) = b.Since b is non-zero, we can divide both sides by b, to get -12 (b + 1) = 1. This leads to b = -3. Then a = 12, so the roots are -2,32.