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Simplifying 9k2 + 12k + 4 = 20 Reorder the terms: 4 + 12k + 9k2 = 20 Solving 4 + 12k + 9k2 = 20 Solving for variable 'k'. Reorder the terms: 4 + -20 + 12k + 9k2 = 20 + -20 Combine like terms: 4 + -20 = -16 -16 + 12k + 9k2 = 20 + -20 Combine like terms: 20 + -20 = 0 -16 + 12k + 9k2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + 1.333333333k + k2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 1.333333333k + 1.777777778 + k2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 1.333333333k + k2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 1.333333333k + k2 = 0 + 1.777777778 1.333333333k + k2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 1.333333333k + k2 = 1.777777778 The k term is 1.333333333k. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333k + 0.4444444442 + k2 = 1.777777778 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333k + k2 = 1.777777778 + 0.4444444442 Combine like terms: 1.777777778 + 0.4444444442 = 2.2222222222 0.4444444442 + 1.333333333k + k2 = 2.2222222222 Factor a perfect square on the left side: (k + 0.6666666665)(k + 0.6666666665) = 2.2222222222 Calculate the square root of the right side: 1.490711985 Break this problem into two subproblems by setting (k + 0.6666666665) equal to 1.490711985 and -1.490711985.Subproblem 1
k + 0.6666666665 = 1.490711985 Simplifying k + 0.6666666665 = 1.490711985 Reorder the terms: 0.6666666665 + k = 1.490711985 Solving 0.6666666665 + k = 1.490711985 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + k = 1.490711985 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + k = 1.490711985 + -0.6666666665 k = 1.490711985 + -0.6666666665 Combine like terms: 1.490711985 + -0.6666666665 = 0.8240453185 k = 0.8240453185 Simplifying k = 0.8240453185Subproblem 2
k + 0.6666666665 = -1.490711985 Simplifying k + 0.6666666665 = -1.490711985 Reorder the terms: 0.6666666665 + k = -1.490711985 Solving 0.6666666665 + k = -1.490711985 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + k = -1.490711985 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + k = -1.490711985 + -0.6666666665 k = -1.490711985 + -0.6666666665 Combine like terms: -1.490711985 + -0.6666666665 = -2.1573786515 k = -2.1573786515 Simplifying k = -2.1573786515Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.8240453185, -2.1573786515}
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