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Simplifying 18 + 2k + -5k2 = 0 Solving 18 + 2k + -5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -3.6 + -0.4k + k2 = 0 Move the constant term to the right: Add '3.6' to each side of the equation. -3.6 + -0.4k + 3.6 + k2 = 0 + 3.6 Reorder the terms: -3.6 + 3.6 + -0.4k + k2 = 0 + 3.6 Combine like terms: -3.6 + 3.6 = 0.0 0.0 + -0.4k + k2 = 0 + 3.6 -0.4k + k2 = 0 + 3.6 Combine like terms: 0 + 3.6 = 3.6 -0.4k + k2 = 3.6 The k term is -0.4k. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4k + 0.04 + k2 = 3.6 + 0.04 Reorder the terms: 0.04 + -0.4k + k2 = 3.6 + 0.04 Combine like terms: 3.6 + 0.04 = 3.64 0.04 + -0.4k + k2 = 3.64 Factor a perfect square on the left side: (k + -0.2)(k + -0.2) = 3.64 Calculate the square root of the right side: 1.907878403 Break this problem into two subproblems by setting (k + -0.2) equal to 1.907878403 and -1.907878403.Subproblem 1
k + -0.2 = 1.907878403 Simplifying k + -0.2 = 1.907878403 Reorder the terms: -0.2 + k = 1.907878403 Solving -0.2 + k = 1.907878403 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + k = 1.907878403 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + k = 1.907878403 + 0.2 k = 1.907878403 + 0.2 Combine like terms: 1.907878403 + 0.2 = 2.107878403 k = 2.107878403 Simplifying k = 2.107878403Subproblem 2
k + -0.2 = -1.907878403 Simplifying k + -0.2 = -1.907878403 Reorder the terms: -0.2 + k = -1.907878403 Solving -0.2 + k = -1.907878403 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + k = -1.907878403 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + k = -1.907878403 + 0.2 k = -1.907878403 + 0.2 Combine like terms: -1.907878403 + 0.2 = -1.707878403 k = -1.707878403 Simplifying k = -1.707878403Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.107878403, -1.707878403}
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