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Simplifying 5k2 + -1(-6k2 + -2k + 6) = 0 Reorder the terms: 5k2 + -1(6 + -2k + -6k2) = 0 5k2 + (6 * -1 + -2k * -1 + -6k2 * -1) = 0 5k2 + (-6 + 2k + 6k2) = 0 Reorder the terms: -6 + 2k + 5k2 + 6k2 = 0 Combine like terms: 5k2 + 6k2 = 11k2 -6 + 2k + 11k2 = 0 Solving -6 + 2k + 11k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.5454545455 + 0.1818181818k + k2 = 0 Move the constant term to the right: Add '0.5454545455' to each side of the equation. -0.5454545455 + 0.1818181818k + 0.5454545455 + k2 = 0 + 0.5454545455 Reorder the terms: -0.5454545455 + 0.5454545455 + 0.1818181818k + k2 = 0 + 0.5454545455 Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000 0.0000000000 + 0.1818181818k + k2 = 0 + 0.5454545455 0.1818181818k + k2 = 0 + 0.5454545455 Combine like terms: 0 + 0.5454545455 = 0.5454545455 0.1818181818k + k2 = 0.5454545455 The k term is 0.1818181818k. Take half its coefficient (0.0909090909). Square it (0.008264462808) and add it to both sides. Add '0.008264462808' to each side of the equation. 0.1818181818k + 0.008264462808 + k2 = 0.5454545455 + 0.008264462808 Reorder the terms: 0.008264462808 + 0.1818181818k + k2 = 0.5454545455 + 0.008264462808 Combine like terms: 0.5454545455 + 0.008264462808 = 0.553719008308 0.008264462808 + 0.1818181818k + k2 = 0.553719008308 Factor a perfect square on the left side: (k + 0.0909090909)(k + 0.0909090909) = 0.553719008308 Calculate the square root of the right side: 0.744122979 Break this problem into two subproblems by setting (k + 0.0909090909) equal to 0.744122979 and -0.744122979.Subproblem 1
k + 0.0909090909 = 0.744122979 Simplifying k + 0.0909090909 = 0.744122979 Reorder the terms: 0.0909090909 + k = 0.744122979 Solving 0.0909090909 + k = 0.744122979 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + k = 0.744122979 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + k = 0.744122979 + -0.0909090909 k = 0.744122979 + -0.0909090909 Combine like terms: 0.744122979 + -0.0909090909 = 0.6532138881 k = 0.6532138881 Simplifying k = 0.6532138881Subproblem 2
k + 0.0909090909 = -0.744122979 Simplifying k + 0.0909090909 = -0.744122979 Reorder the terms: 0.0909090909 + k = -0.744122979 Solving 0.0909090909 + k = -0.744122979 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + k = -0.744122979 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + k = -0.744122979 + -0.0909090909 k = -0.744122979 + -0.0909090909 Combine like terms: -0.744122979 + -0.0909090909 = -0.8350320699 k = -0.8350320699 Simplifying k = -0.8350320699Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.6532138881, -0.8350320699}
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