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Simplifying 6k2 + -22k + 8 = 0 Reorder the terms: 8 + -22k + 6k2 = 0 Solving 8 + -22k + 6k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(4 + -11k + 3k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + -11k + 3k2)' equal to zero and attempt to solve: Simplifying 4 + -11k + 3k2 = 0 Solving 4 + -11k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + -3.666666667k + k2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -3.666666667k + -1.333333333 + k2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -3.666666667k + k2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -3.666666667k + k2 = 0 + -1.333333333 -3.666666667k + k2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -3.666666667k + k2 = -1.333333333 The k term is -3.666666667k. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667k + 3.361111114 + k2 = -1.333333333 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667k + k2 = -1.333333333 + 3.361111114 Combine like terms: -1.333333333 + 3.361111114 = 2.027777781 3.361111114 + -3.666666667k + k2 = 2.027777781 Factor a perfect square on the left side: (k + -1.833333334)(k + -1.833333334) = 2.027777781 Calculate the square root of the right side: 1.424000625 Break this problem into two subproblems by setting (k + -1.833333334) equal to 1.424000625 and -1.424000625.Subproblem 1
k + -1.833333334 = 1.424000625 Simplifying k + -1.833333334 = 1.424000625 Reorder the terms: -1.833333334 + k = 1.424000625 Solving -1.833333334 + k = 1.424000625 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + k = 1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + k = 1.424000625 + 1.833333334 k = 1.424000625 + 1.833333334 Combine like terms: 1.424000625 + 1.833333334 = 3.257333959 k = 3.257333959 Simplifying k = 3.257333959Subproblem 2
k + -1.833333334 = -1.424000625 Simplifying k + -1.833333334 = -1.424000625 Reorder the terms: -1.833333334 + k = -1.424000625 Solving -1.833333334 + k = -1.424000625 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + k = -1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + k = -1.424000625 + 1.833333334 k = -1.424000625 + 1.833333334 Combine like terms: -1.424000625 + 1.833333334 = 0.409332709 k = 0.409332709 Simplifying k = 0.409332709Solution
The solution to the problem is based on the solutions from the subproblems. k = {3.257333959, 0.409332709}Solution
k = {3.257333959, 0.409332709}
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