If α and β are roots of the equation x2 – x – 1 = 0, then the equation whose roots are α/β and β/α is:
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x2 + 3x – 1 = 0
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x2 + x – 1 = 0
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x2 – x + 1 = 0
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x2 + 3x + 1 = 0
Roots are a, b. a+b = 1, ab = -1. New roots: a/b, b/a. Sum = (a^2+b^2)/ab = ((a+b)^2 - 2ab)/ab = (1 - 2(-1)) / -1 = 3 / -1 = -3. Product = (a/b)*(b/a) = 1. Equation: x^2 - (sum)x + product = 0 => x^2 - (-3)x + 1 = 0 => x^2 + 3x + 1 = 0.
Using Vieta's formulas for the equation x squared minus x minus 1 equals 0, the sum of the roots alpha plus beta is 1 and the product alpha times beta is negative 1. For the new roots alpha divided by beta and beta divided by alpha, the sum is (alpha squared plus beta squared) divided by (alpha beta), which simplifies to ((alpha plus beta) squared minus 2(alpha beta)) divided by (alpha beta), yielding negative 3. The product of the new roots is (alpha beta) divided by (alpha beta) which equals 1. The required quadratic equation is x squared minus the sum of the roots times x plus the product of the roots equals 0, resulting in x squared plus 3x plus 1 equals 0.