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Simplifying 6k2 + 12k + -52 = 0 Reorder the terms: -52 + 12k + 6k2 = 0 Solving -52 + 12k + 6k2 = 0 Solving for variable 'k'. Factor out the Greatest Common Factor (GCF), '2'. 2(-26 + 6k + 3k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-26 + 6k + 3k2)' equal to zero and attempt to solve: Simplifying -26 + 6k + 3k2 = 0 Solving -26 + 6k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -8.666666667 + 2k + k2 = 0 Move the constant term to the right: Add '8.666666667' to each side of the equation. -8.666666667 + 2k + 8.666666667 + k2 = 0 + 8.666666667 Reorder the terms: -8.666666667 + 8.666666667 + 2k + k2 = 0 + 8.666666667 Combine like terms: -8.666666667 + 8.666666667 = 0.000000000 0.000000000 + 2k + k2 = 0 + 8.666666667 2k + k2 = 0 + 8.666666667 Combine like terms: 0 + 8.666666667 = 8.666666667 2k + k2 = 8.666666667 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 8.666666667 + 1 Reorder the terms: 1 + 2k + k2 = 8.666666667 + 1 Combine like terms: 8.666666667 + 1 = 9.666666667 1 + 2k + k2 = 9.666666667 Factor a perfect square on the left side: (k + 1)(k + 1) = 9.666666667 Calculate the square root of the right side: 3.109126351 Break this problem into two subproblems by setting (k + 1) equal to 3.109126351 and -3.109126351.Subproblem 1
k + 1 = 3.109126351 Simplifying k + 1 = 3.109126351 Reorder the terms: 1 + k = 3.109126351 Solving 1 + k = 3.109126351 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 3.109126351 + -1 Combine like terms: 1 + -1 = 0 0 + k = 3.109126351 + -1 k = 3.109126351 + -1 Combine like terms: 3.109126351 + -1 = 2.109126351 k = 2.109126351 Simplifying k = 2.109126351Subproblem 2
k + 1 = -3.109126351 Simplifying k + 1 = -3.109126351 Reorder the terms: 1 + k = -3.109126351 Solving 1 + k = -3.109126351 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -3.109126351 + -1 Combine like terms: 1 + -1 = 0 0 + k = -3.109126351 + -1 k = -3.109126351 + -1 Combine like terms: -3.109126351 + -1 = -4.109126351 k = -4.109126351 Simplifying k = -4.109126351Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.109126351, -4.109126351}Solution
k = {2.109126351, -4.109126351}
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