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Simplifying 3k2 + 12k + 4 = 0 Reorder the terms: 4 + 12k + 3k2 = 0 Solving 4 + 12k + 3k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 4k + k2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 4k + -1.333333333 + k2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 4k + k2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 4k + k2 = 0 + -1.333333333 4k + k2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 4k + k2 = -1.333333333 The k term is 4k. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4k + 4 + k2 = -1.333333333 + 4 Reorder the terms: 4 + 4k + k2 = -1.333333333 + 4 Combine like terms: -1.333333333 + 4 = 2.666666667 4 + 4k + k2 = 2.666666667 Factor a perfect square on the left side: (k + 2)(k + 2) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (k + 2) equal to 1.632993162 and -1.632993162.Subproblem 1
k + 2 = 1.632993162 Simplifying k + 2 = 1.632993162 Reorder the terms: 2 + k = 1.632993162 Solving 2 + k = 1.632993162 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + k = 1.632993162 + -2 Combine like terms: 2 + -2 = 0 0 + k = 1.632993162 + -2 k = 1.632993162 + -2 Combine like terms: 1.632993162 + -2 = -0.367006838 k = -0.367006838 Simplifying k = -0.367006838Subproblem 2
k + 2 = -1.632993162 Simplifying k + 2 = -1.632993162 Reorder the terms: 2 + k = -1.632993162 Solving 2 + k = -1.632993162 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + k = -1.632993162 + -2 Combine like terms: 2 + -2 = 0 0 + k = -1.632993162 + -2 k = -1.632993162 + -2 Combine like terms: -1.632993162 + -2 = -3.632993162 k = -3.632993162 Simplifying k = -3.632993162Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.367006838, -3.632993162}
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