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Simplifying 12k2 + 1 = 12k Reorder the terms: 1 + 12k2 = 12k Solving 1 + 12k2 = 12k Solving for variable 'k'. Reorder the terms: 1 + -12k + 12k2 = 12k + -12k Combine like terms: 12k + -12k = 0 1 + -12k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.08333333333 + -1k + k2 = 0 Move the constant term to the right: Add '-0.08333333333' to each side of the equation. 0.08333333333 + -1k + -0.08333333333 + k2 = 0 + -0.08333333333 Reorder the terms: 0.08333333333 + -0.08333333333 + -1k + k2 = 0 + -0.08333333333 Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000 0.00000000000 + -1k + k2 = 0 + -0.08333333333 -1k + k2 = 0 + -0.08333333333 Combine like terms: 0 + -0.08333333333 = -0.08333333333 -1k + k2 = -0.08333333333 The k term is -1k. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1k + 0.25 + k2 = -0.08333333333 + 0.25 Reorder the terms: 0.25 + -1k + k2 = -0.08333333333 + 0.25 Combine like terms: -0.08333333333 + 0.25 = 0.16666666667 0.25 + -1k + k2 = 0.16666666667 Factor a perfect square on the left side: (k + -0.5)(k + -0.5) = 0.16666666667 Calculate the square root of the right side: 0.40824829 Break this problem into two subproblems by setting (k + -0.5) equal to 0.40824829 and -0.40824829.Subproblem 1
k + -0.5 = 0.40824829 Simplifying k + -0.5 = 0.40824829 Reorder the terms: -0.5 + k = 0.40824829 Solving -0.5 + k = 0.40824829 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = 0.40824829 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = 0.40824829 + 0.5 k = 0.40824829 + 0.5 Combine like terms: 0.40824829 + 0.5 = 0.90824829 k = 0.90824829 Simplifying k = 0.90824829Subproblem 2
k + -0.5 = -0.40824829 Simplifying k + -0.5 = -0.40824829 Reorder the terms: -0.5 + k = -0.40824829 Solving -0.5 + k = -0.40824829 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k = -0.40824829 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k = -0.40824829 + 0.5 k = -0.40824829 + 0.5 Combine like terms: -0.40824829 + 0.5 = 0.09175171 k = 0.09175171 Simplifying k = 0.09175171Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.90824829, 0.09175171}
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