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Simplifying 18 = 2(3k + 1) * k Reorder the terms: 18 = 2(1 + 3k) * k Reorder the terms for easier multiplication: 18 = 2k(1 + 3k) 18 = (1 * 2k + 3k * 2k) 18 = (2k + 6k2) Solving 18 = 2k + 6k2 Solving for variable 'k'. Reorder the terms: 18 + -2k + -6k2 = 2k + -2k + 6k2 + -6k2 Combine like terms: 2k + -2k = 0 18 + -2k + -6k2 = 0 + 6k2 + -6k2 18 + -2k + -6k2 = 6k2 + -6k2 Combine like terms: 6k2 + -6k2 = 0 18 + -2k + -6k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(9 + -1k + -3k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(9 + -1k + -3k2)' equal to zero and attempt to solve: Simplifying 9 + -1k + -3k2 = 0 Solving 9 + -1k + -3k2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -3 + 0.3333333333k + k2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.3333333333k + 3 + k2 = 0 + 3 Reorder the terms: -3 + 3 + 0.3333333333k + k2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.3333333333k + k2 = 0 + 3 0.3333333333k + k2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.3333333333k + k2 = 3 The k term is 0.3333333333k. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333k + 0.02777777779 + k2 = 3 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333k + k2 = 3 + 0.02777777779 Combine like terms: 3 + 0.02777777779 = 3.02777777779 0.02777777779 + 0.3333333333k + k2 = 3.02777777779 Factor a perfect square on the left side: (k + 0.1666666667)(k + 0.1666666667) = 3.02777777779 Calculate the square root of the right side: 1.740051085 Break this problem into two subproblems by setting (k + 0.1666666667) equal to 1.740051085 and -1.740051085.Subproblem 1
k + 0.1666666667 = 1.740051085 Simplifying k + 0.1666666667 = 1.740051085 Reorder the terms: 0.1666666667 + k = 1.740051085 Solving 0.1666666667 + k = 1.740051085 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + k = 1.740051085 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + k = 1.740051085 + -0.1666666667 k = 1.740051085 + -0.1666666667 Combine like terms: 1.740051085 + -0.1666666667 = 1.5733844183 k = 1.5733844183 Simplifying k = 1.5733844183Subproblem 2
k + 0.1666666667 = -1.740051085 Simplifying k + 0.1666666667 = -1.740051085 Reorder the terms: 0.1666666667 + k = -1.740051085 Solving 0.1666666667 + k = -1.740051085 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + k = -1.740051085 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + k = -1.740051085 + -0.1666666667 k = -1.740051085 + -0.1666666667 Combine like terms: -1.740051085 + -0.1666666667 = -1.9067177517 k = -1.9067177517 Simplifying k = -1.9067177517Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.5733844183, -1.9067177517}Solution
k = {1.5733844183, -1.9067177517}
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