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Simplifying 3k2 + 6k + -1 = 4 Reorder the terms: -1 + 6k + 3k2 = 4 Solving -1 + 6k + 3k2 = 4 Solving for variable 'k'. Reorder the terms: -1 + -4 + 6k + 3k2 = 4 + -4 Combine like terms: -1 + -4 = -5 -5 + 6k + 3k2 = 4 + -4 Combine like terms: 4 + -4 = 0 -5 + 6k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + 2k + k2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 2k + 1.666666667 + k2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 2k + k2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 2k + k2 = 0 + 1.666666667 2k + k2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 2k + k2 = 1.666666667 The k term is 2k. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2k + 1 + k2 = 1.666666667 + 1 Reorder the terms: 1 + 2k + k2 = 1.666666667 + 1 Combine like terms: 1.666666667 + 1 = 2.666666667 1 + 2k + k2 = 2.666666667 Factor a perfect square on the left side: (k + 1)(k + 1) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (k + 1) equal to 1.632993162 and -1.632993162.Subproblem 1
k + 1 = 1.632993162 Simplifying k + 1 = 1.632993162 Reorder the terms: 1 + k = 1.632993162 Solving 1 + k = 1.632993162 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = 1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + k = 1.632993162 + -1 k = 1.632993162 + -1 Combine like terms: 1.632993162 + -1 = 0.632993162 k = 0.632993162 Simplifying k = 0.632993162Subproblem 2
k + 1 = -1.632993162 Simplifying k + 1 = -1.632993162 Reorder the terms: 1 + k = -1.632993162 Solving 1 + k = -1.632993162 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + k = -1.632993162 + -1 Combine like terms: 1 + -1 = 0 0 + k = -1.632993162 + -1 k = -1.632993162 + -1 Combine like terms: -1.632993162 + -1 = -2.632993162 k = -2.632993162 Simplifying k = -2.632993162Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.632993162, -2.632993162}
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