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Simplifying 11k2 = 3 + -3k Solving 11k2 = 3 + -3k Solving for variable 'k'. Reorder the terms: -3 + 3k + 11k2 = 3 + -3k + -3 + 3k Reorder the terms: -3 + 3k + 11k2 = 3 + -3 + -3k + 3k Combine like terms: 3 + -3 = 0 -3 + 3k + 11k2 = 0 + -3k + 3k -3 + 3k + 11k2 = -3k + 3k Combine like terms: -3k + 3k = 0 -3 + 3k + 11k2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.2727272727 + 0.2727272727k + k2 = 0 Move the constant term to the right: Add '0.2727272727' to each side of the equation. -0.2727272727 + 0.2727272727k + 0.2727272727 + k2 = 0 + 0.2727272727 Reorder the terms: -0.2727272727 + 0.2727272727 + 0.2727272727k + k2 = 0 + 0.2727272727 Combine like terms: -0.2727272727 + 0.2727272727 = 0.0000000000 0.0000000000 + 0.2727272727k + k2 = 0 + 0.2727272727 0.2727272727k + k2 = 0 + 0.2727272727 Combine like terms: 0 + 0.2727272727 = 0.2727272727 0.2727272727k + k2 = 0.2727272727 The k term is 0.2727272727k. Take half its coefficient (0.1363636364). Square it (0.01859504133) and add it to both sides. Add '0.01859504133' to each side of the equation. 0.2727272727k + 0.01859504133 + k2 = 0.2727272727 + 0.01859504133 Reorder the terms: 0.01859504133 + 0.2727272727k + k2 = 0.2727272727 + 0.01859504133 Combine like terms: 0.2727272727 + 0.01859504133 = 0.29132231403 0.01859504133 + 0.2727272727k + k2 = 0.29132231403 Factor a perfect square on the left side: (k + 0.1363636364)(k + 0.1363636364) = 0.29132231403 Calculate the square root of the right side: 0.539742822 Break this problem into two subproblems by setting (k + 0.1363636364) equal to 0.539742822 and -0.539742822.Subproblem 1
k + 0.1363636364 = 0.539742822 Simplifying k + 0.1363636364 = 0.539742822 Reorder the terms: 0.1363636364 + k = 0.539742822 Solving 0.1363636364 + k = 0.539742822 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + k = 0.539742822 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + k = 0.539742822 + -0.1363636364 k = 0.539742822 + -0.1363636364 Combine like terms: 0.539742822 + -0.1363636364 = 0.4033791856 k = 0.4033791856 Simplifying k = 0.4033791856Subproblem 2
k + 0.1363636364 = -0.539742822 Simplifying k + 0.1363636364 = -0.539742822 Reorder the terms: 0.1363636364 + k = -0.539742822 Solving 0.1363636364 + k = -0.539742822 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + k = -0.539742822 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + k = -0.539742822 + -0.1363636364 k = -0.539742822 + -0.1363636364 Combine like terms: -0.539742822 + -0.1363636364 = -0.6761064584 k = -0.6761064584 Simplifying k = -0.6761064584Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.4033791856, -0.6761064584}
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