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Simplifying 12k2 + 15k = 6k + 20 Reorder the terms: 15k + 12k2 = 6k + 20 Reorder the terms: 15k + 12k2 = 20 + 6k Solving 15k + 12k2 = 20 + 6k Solving for variable 'k'. Reorder the terms: -20 + 15k + -6k + 12k2 = 20 + 6k + -20 + -6k Combine like terms: 15k + -6k = 9k -20 + 9k + 12k2 = 20 + 6k + -20 + -6k Reorder the terms: -20 + 9k + 12k2 = 20 + -20 + 6k + -6k Combine like terms: 20 + -20 = 0 -20 + 9k + 12k2 = 0 + 6k + -6k -20 + 9k + 12k2 = 6k + -6k Combine like terms: 6k + -6k = 0 -20 + 9k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.666666667 + 0.75k + k2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 0.75k + 1.666666667 + k2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 0.75k + k2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 0.75k + k2 = 0 + 1.666666667 0.75k + k2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 0.75k + k2 = 1.666666667 The k term is 0.75k. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75k + 0.140625 + k2 = 1.666666667 + 0.140625 Reorder the terms: 0.140625 + 0.75k + k2 = 1.666666667 + 0.140625 Combine like terms: 1.666666667 + 0.140625 = 1.807291667 0.140625 + 0.75k + k2 = 1.807291667 Factor a perfect square on the left side: (k + 0.375)(k + 0.375) = 1.807291667 Calculate the square root of the right side: 1.344355484 Break this problem into two subproblems by setting (k + 0.375) equal to 1.344355484 and -1.344355484.Subproblem 1
k + 0.375 = 1.344355484 Simplifying k + 0.375 = 1.344355484 Reorder the terms: 0.375 + k = 1.344355484 Solving 0.375 + k = 1.344355484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = 1.344355484 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = 1.344355484 + -0.375 k = 1.344355484 + -0.375 Combine like terms: 1.344355484 + -0.375 = 0.969355484 k = 0.969355484 Simplifying k = 0.969355484Subproblem 2
k + 0.375 = -1.344355484 Simplifying k + 0.375 = -1.344355484 Reorder the terms: 0.375 + k = -1.344355484 Solving 0.375 + k = -1.344355484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = -1.344355484 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = -1.344355484 + -0.375 k = -1.344355484 + -0.375 Combine like terms: -1.344355484 + -0.375 = -1.719355484 k = -1.719355484 Simplifying k = -1.719355484Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.969355484, -1.719355484}
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