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Simplifying -2k2 + 6k + -3 = 0 Reorder the terms: -3 + 6k + -2k2 = 0 Solving -3 + 6k + -2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 1.5 + -3k + k2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3k + -1.5 + k2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3k + k2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3k + k2 = 0 + -1.5 -3k + k2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3k + k2 = -1.5 The k term is -3k. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3k + 2.25 + k2 = -1.5 + 2.25 Reorder the terms: 2.25 + -3k + k2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3k + k2 = 0.75 Factor a perfect square on the left side: (k + -1.5)(k + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (k + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
k + -1.5 = 0.866025404 Simplifying k + -1.5 = 0.866025404 Reorder the terms: -1.5 + k = 0.866025404 Solving -1.5 + k = 0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = 0.866025404 + 1.5 k = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 k = 2.366025404 Simplifying k = 2.366025404Subproblem 2
k + -1.5 = -0.866025404 Simplifying k + -1.5 = -0.866025404 Reorder the terms: -1.5 + k = -0.866025404 Solving -1.5 + k = -0.866025404 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = -0.866025404 + 1.5 k = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 k = 0.633974596 Simplifying k = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.366025404, 0.633974596}
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