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Simplifying 13k2 + 36k + -24 = 0 Reorder the terms: -24 + 36k + 13k2 = 0 Solving -24 + 36k + 13k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -1.846153846 + 2.769230769k + k2 = 0 Move the constant term to the right: Add '1.846153846' to each side of the equation. -1.846153846 + 2.769230769k + 1.846153846 + k2 = 0 + 1.846153846 Reorder the terms: -1.846153846 + 1.846153846 + 2.769230769k + k2 = 0 + 1.846153846 Combine like terms: -1.846153846 + 1.846153846 = 0.000000000 0.000000000 + 2.769230769k + k2 = 0 + 1.846153846 2.769230769k + k2 = 0 + 1.846153846 Combine like terms: 0 + 1.846153846 = 1.846153846 2.769230769k + k2 = 1.846153846 The k term is 2.769230769k. Take half its coefficient (1.384615385). Square it (1.917159764) and add it to both sides. Add '1.917159764' to each side of the equation. 2.769230769k + 1.917159764 + k2 = 1.846153846 + 1.917159764 Reorder the terms: 1.917159764 + 2.769230769k + k2 = 1.846153846 + 1.917159764 Combine like terms: 1.846153846 + 1.917159764 = 3.76331361 1.917159764 + 2.769230769k + k2 = 3.76331361 Factor a perfect square on the left side: (k + 1.384615385)(k + 1.384615385) = 3.76331361 Calculate the square root of the right side: 1.939926187 Break this problem into two subproblems by setting (k + 1.384615385) equal to 1.939926187 and -1.939926187.Subproblem 1
k + 1.384615385 = 1.939926187 Simplifying k + 1.384615385 = 1.939926187 Reorder the terms: 1.384615385 + k = 1.939926187 Solving 1.384615385 + k = 1.939926187 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.384615385' to each side of the equation. 1.384615385 + -1.384615385 + k = 1.939926187 + -1.384615385 Combine like terms: 1.384615385 + -1.384615385 = 0.000000000 0.000000000 + k = 1.939926187 + -1.384615385 k = 1.939926187 + -1.384615385 Combine like terms: 1.939926187 + -1.384615385 = 0.555310802 k = 0.555310802 Simplifying k = 0.555310802Subproblem 2
k + 1.384615385 = -1.939926187 Simplifying k + 1.384615385 = -1.939926187 Reorder the terms: 1.384615385 + k = -1.939926187 Solving 1.384615385 + k = -1.939926187 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.384615385' to each side of the equation. 1.384615385 + -1.384615385 + k = -1.939926187 + -1.384615385 Combine like terms: 1.384615385 + -1.384615385 = 0.000000000 0.000000000 + k = -1.939926187 + -1.384615385 k = -1.939926187 + -1.384615385 Combine like terms: -1.939926187 + -1.384615385 = -3.324541572 k = -3.324541572 Simplifying k = -3.324541572Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.555310802, -3.324541572}
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