Four equations are given below.

Given the quadratic equation: x^2 - x + 1 = 0.

Step-by-Step Solution

1. Use the relation between roots and coefficients: For an equation ax^2 + bx + c = 0, the sum of roots is (alpha + beta) = -b/a and the product of roots is (alpha * beta) = c/a. Here, a = 1, b = -1, c = 1.

   • Sum of roots: alpha + beta = -(-1)/1 = 1

   • Product of roots: alpha * beta = 1/1 = 1

2. Find the sum and product of the new roots (alpha^3 and beta^3):

   • New sum of roots = alpha^3 + beta^3 Using the formula: a^3 + b^3 = (a + b)^3 - 3ab(a + b) alpha^3 + beta^3 = (alpha + beta)^3 - 3(alpha * beta)(alpha + beta) alpha^3 + beta^3 = (1)^3 - 3(1)(1) = 1 - 3 = -2

   • New product of roots = alpha^3 * beta^3 = (alpha * beta)^3 alpha^3 * beta^3 = (1)^3 = 1

3. Form the new equation: The general form for a quadratic equation with sum of roots S and product of roots P is: x^2 - Sx + P = 0. Substituting S = -2 and P = 1: x^2 - (-2)x + 1 = 0 x^2 + 2x + 1 = 0