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Simplifying k2 + -3k + -5 = 0 Reorder the terms: -5 + -3k + k2 = 0 Solving -5 + -3k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + -3k + 5 + k2 = 0 + 5 Reorder the terms: -5 + 5 + -3k + k2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3k + k2 = 0 + 5 -3k + k2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3k + k2 = 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 = 5 + 2.25 Reorder the terms: 2.25 + -3k + k2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + -3k + k2 = 7.25 Factor a perfect square on the left side: (k + -1.5)(k + -1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (k + -1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
k + -1.5 = 2.692582404 Simplifying k + -1.5 = 2.692582404 Reorder the terms: -1.5 + k = 2.692582404 Solving -1.5 + k = 2.692582404 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 = 2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = 2.692582404 + 1.5 k = 2.692582404 + 1.5 Combine like terms: 2.692582404 + 1.5 = 4.192582404 k = 4.192582404 Simplifying k = 4.192582404Subproblem 2
k + -1.5 = -2.692582404 Simplifying k + -1.5 = -2.692582404 Reorder the terms: -1.5 + k = -2.692582404 Solving -1.5 + k = -2.692582404 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 = -2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = -2.692582404 + 1.5 k = -2.692582404 + 1.5 Combine like terms: -2.692582404 + 1.5 = -1.192582404 k = -1.192582404 Simplifying k = -1.192582404Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.192582404, -1.192582404}
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