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Simplifying 40k2 + -3k + -0.3 = 0 Reorder the terms: -0.3 + -3k + 40k2 = 0 Solving -0.3 + -3k + 40k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.0075 + -0.075k + k2 = 0 Move the constant term to the right: Add '0.0075' to each side of the equation. -0.0075 + -0.075k + 0.0075 + k2 = 0 + 0.0075 Reorder the terms: -0.0075 + 0.0075 + -0.075k + k2 = 0 + 0.0075 Combine like terms: -0.0075 + 0.0075 = 0.0000 0.0000 + -0.075k + k2 = 0 + 0.0075 -0.075k + k2 = 0 + 0.0075 Combine like terms: 0 + 0.0075 = 0.0075 -0.075k + k2 = 0.0075 The k term is -0.075k. Take half its coefficient (-0.0375). Square it (0.00140625) and add it to both sides. Add '0.00140625' to each side of the equation. -0.075k + 0.00140625 + k2 = 0.0075 + 0.00140625 Reorder the terms: 0.00140625 + -0.075k + k2 = 0.0075 + 0.00140625 Combine like terms: 0.0075 + 0.00140625 = 0.00890625 0.00140625 + -0.075k + k2 = 0.00890625 Factor a perfect square on the left side: (k + -0.0375)(k + -0.0375) = 0.00890625 Calculate the square root of the right side: 0.09437293 Break this problem into two subproblems by setting (k + -0.0375) equal to 0.09437293 and -0.09437293.Subproblem 1
k + -0.0375 = 0.09437293 Simplifying k + -0.0375 = 0.09437293 Reorder the terms: -0.0375 + k = 0.09437293 Solving -0.0375 + k = 0.09437293 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.0375' to each side of the equation. -0.0375 + 0.0375 + k = 0.09437293 + 0.0375 Combine like terms: -0.0375 + 0.0375 = 0.0000 0.0000 + k = 0.09437293 + 0.0375 k = 0.09437293 + 0.0375 Combine like terms: 0.09437293 + 0.0375 = 0.13187293 k = 0.13187293 Simplifying k = 0.13187293Subproblem 2
k + -0.0375 = -0.09437293 Simplifying k + -0.0375 = -0.09437293 Reorder the terms: -0.0375 + k = -0.09437293 Solving -0.0375 + k = -0.09437293 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.0375' to each side of the equation. -0.0375 + 0.0375 + k = -0.09437293 + 0.0375 Combine like terms: -0.0375 + 0.0375 = 0.0000 0.0000 + k = -0.09437293 + 0.0375 k = -0.09437293 + 0.0375 Combine like terms: -0.09437293 + 0.0375 = -0.05687293 k = -0.05687293 Simplifying k = -0.05687293Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.13187293, -0.05687293}
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