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Simplifying 12k2 + -9k + -2 = 3 Reorder the terms: -2 + -9k + 12k2 = 3 Solving -2 + -9k + 12k2 = 3 Solving for variable 'k'. Reorder the terms: -2 + -3 + -9k + 12k2 = 3 + -3 Combine like terms: -2 + -3 = -5 -5 + -9k + 12k2 = 3 + -3 Combine like terms: 3 + -3 = 0 -5 + -9k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -0.75k + k2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -0.75k + 0.4166666667 + k2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -0.75k + k2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -0.75k + k2 = 0 + 0.4166666667 -0.75k + k2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -0.75k + k2 = 0.4166666667 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 = 0.4166666667 + 0.140625 Reorder the terms: 0.140625 + -0.75k + k2 = 0.4166666667 + 0.140625 Combine like terms: 0.4166666667 + 0.140625 = 0.5572916667 0.140625 + -0.75k + k2 = 0.5572916667 Factor a perfect square on the left side: (k + -0.375)(k + -0.375) = 0.5572916667 Calculate the square root of the right side: 0.746519703 Break this problem into two subproblems by setting (k + -0.375) equal to 0.746519703 and -0.746519703.Subproblem 1
k + -0.375 = 0.746519703 Simplifying k + -0.375 = 0.746519703 Reorder the terms: -0.375 + k = 0.746519703 Solving -0.375 + k = 0.746519703 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 = 0.746519703 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + k = 0.746519703 + 0.375 k = 0.746519703 + 0.375 Combine like terms: 0.746519703 + 0.375 = 1.121519703 k = 1.121519703 Simplifying k = 1.121519703Subproblem 2
k + -0.375 = -0.746519703 Simplifying k + -0.375 = -0.746519703 Reorder the terms: -0.375 + k = -0.746519703 Solving -0.375 + k = -0.746519703 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 = -0.746519703 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + k = -0.746519703 + 0.375 k = -0.746519703 + 0.375 Combine like terms: -0.746519703 + 0.375 = -0.371519703 k = -0.371519703 Simplifying k = -0.371519703Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.121519703, -0.371519703}
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