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Simplifying -1 = 4k2 + -6k Reorder the terms: -1 = -6k + 4k2 Solving -1 = -6k + 4k2 Solving for variable 'k'. Reorder the terms: -1 + 6k + -4k2 = -6k + 6k + 4k2 + -4k2 Combine like terms: -6k + 6k = 0 -1 + 6k + -4k2 = 0 + 4k2 + -4k2 -1 + 6k + -4k2 = 4k2 + -4k2 Combine like terms: 4k2 + -4k2 = 0 -1 + 6k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.25 + -1.5k + k2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.5k + -0.25 + k2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.5k + k2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.5k + k2 = 0 + -0.25 -1.5k + k2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.5k + k2 = -0.25 The k term is -1.5k. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5k + 0.5625 + k2 = -0.25 + 0.5625 Reorder the terms: 0.5625 + -1.5k + k2 = -0.25 + 0.5625 Combine like terms: -0.25 + 0.5625 = 0.3125 0.5625 + -1.5k + k2 = 0.3125 Factor a perfect square on the left side: (k + -0.75)(k + -0.75) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (k + -0.75) equal to 0.559016994 and -0.559016994.Subproblem 1
k + -0.75 = 0.559016994 Simplifying k + -0.75 = 0.559016994 Reorder the terms: -0.75 + k = 0.559016994 Solving -0.75 + k = 0.559016994 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = 0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = 0.559016994 + 0.75 k = 0.559016994 + 0.75 Combine like terms: 0.559016994 + 0.75 = 1.309016994 k = 1.309016994 Simplifying k = 1.309016994Subproblem 2
k + -0.75 = -0.559016994 Simplifying k + -0.75 = -0.559016994 Reorder the terms: -0.75 + k = -0.559016994 Solving -0.75 + k = -0.559016994 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + k = -0.559016994 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + k = -0.559016994 + 0.75 k = -0.559016994 + 0.75 Combine like terms: -0.559016994 + 0.75 = 0.190983006 k = 0.190983006 Simplifying k = 0.190983006Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.309016994, 0.190983006}
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