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Simplifying 11k2 + 3k + -20 + 0 = 0 Reorder the terms: -20 + 0 + 3k + 11k2 = 0 Combine like terms: -20 + 0 = -20 -20 + 3k + 11k2 = 0 Solving -20 + 3k + 11k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.818181818 + 0.2727272727k + k2 = 0 Move the constant term to the right: Add '1.818181818' to each side of the equation. -1.818181818 + 0.2727272727k + 1.818181818 + k2 = 0 + 1.818181818 Reorder the terms: -1.818181818 + 1.818181818 + 0.2727272727k + k2 = 0 + 1.818181818 Combine like terms: -1.818181818 + 1.818181818 = 0.000000000 0.000000000 + 0.2727272727k + k2 = 0 + 1.818181818 0.2727272727k + k2 = 0 + 1.818181818 Combine like terms: 0 + 1.818181818 = 1.818181818 0.2727272727k + k2 = 1.818181818 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 = 1.818181818 + 0.01859504133 Reorder the terms: 0.01859504133 + 0.2727272727k + k2 = 1.818181818 + 0.01859504133 Combine like terms: 1.818181818 + 0.01859504133 = 1.83677685933 0.01859504133 + 0.2727272727k + k2 = 1.83677685933 Factor a perfect square on the left side: (k + 0.1363636364)(k + 0.1363636364) = 1.83677685933 Calculate the square root of the right side: 1.35527741 Break this problem into two subproblems by setting (k + 0.1363636364) equal to 1.35527741 and -1.35527741.Subproblem 1
k + 0.1363636364 = 1.35527741 Simplifying k + 0.1363636364 = 1.35527741 Reorder the terms: 0.1363636364 + k = 1.35527741 Solving 0.1363636364 + k = 1.35527741 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 = 1.35527741 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + k = 1.35527741 + -0.1363636364 k = 1.35527741 + -0.1363636364 Combine like terms: 1.35527741 + -0.1363636364 = 1.2189137736 k = 1.2189137736 Simplifying k = 1.2189137736Subproblem 2
k + 0.1363636364 = -1.35527741 Simplifying k + 0.1363636364 = -1.35527741 Reorder the terms: 0.1363636364 + k = -1.35527741 Solving 0.1363636364 + k = -1.35527741 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 = -1.35527741 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + k = -1.35527741 + -0.1363636364 k = -1.35527741 + -0.1363636364 Combine like terms: -1.35527741 + -0.1363636364 = -1.4916410464 k = -1.4916410464 Simplifying k = -1.4916410464Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.2189137736, -1.4916410464}
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