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Simplifying -5k2 + -2k + 1 = 0 Reorder the terms: 1 + -2k + -5k2 = 0 Solving 1 + -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'. -0.2 + 0.4k + k2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.4k + 0.2 + k2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.4k + k2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.4k + k2 = 0 + 0.2 0.4k + k2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 0.4k + k2 = 0.2 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 = 0.2 + 0.04 Reorder the terms: 0.04 + 0.4k + k2 = 0.2 + 0.04 Combine like terms: 0.2 + 0.04 = 0.24 0.04 + 0.4k + k2 = 0.24 Factor a perfect square on the left side: (k + 0.2)(k + 0.2) = 0.24 Calculate the square root of the right side: 0.489897949 Break this problem into two subproblems by setting (k + 0.2) equal to 0.489897949 and -0.489897949.Subproblem 1
k + 0.2 = 0.489897949 Simplifying k + 0.2 = 0.489897949 Reorder the terms: 0.2 + k = 0.489897949 Solving 0.2 + k = 0.489897949 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 = 0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + k = 0.489897949 + -0.2 k = 0.489897949 + -0.2 Combine like terms: 0.489897949 + -0.2 = 0.289897949 k = 0.289897949 Simplifying k = 0.289897949Subproblem 2
k + 0.2 = -0.489897949 Simplifying k + 0.2 = -0.489897949 Reorder the terms: 0.2 + k = -0.489897949 Solving 0.2 + k = -0.489897949 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 = -0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + k = -0.489897949 + -0.2 k = -0.489897949 + -0.2 Combine like terms: -0.489897949 + -0.2 = -0.689897949 k = -0.689897949 Simplifying k = -0.689897949Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.289897949, -0.689897949}
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