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Simplifying 2k + 3k(k + 4) = -3 Reorder the terms: 2k + 3k(4 + k) = -3 2k + (4 * 3k + k * 3k) = -3 2k + (12k + 3k2) = -3 Combine like terms: 2k + 12k = 14k 14k + 3k2 = -3 Solving 14k + 3k2 = -3 Solving for variable 'k'. Reorder the terms: 3 + 14k + 3k2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + 14k + 3k2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1 + 4.666666667k + k2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4.666666667k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + 4.666666667k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4.666666667k + k2 = 0 + -1 4.666666667k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 4.666666667k + k2 = -1 The k term is 4.666666667k. Take half its coefficient (2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. 4.666666667k + 5.444444448 + k2 = -1 + 5.444444448 Reorder the terms: 5.444444448 + 4.666666667k + k2 = -1 + 5.444444448 Combine like terms: -1 + 5.444444448 = 4.444444448 5.444444448 + 4.666666667k + k2 = 4.444444448 Factor a perfect square on the left side: (k + 2.333333334)(k + 2.333333334) = 4.444444448 Calculate the square root of the right side: 2.108185108 Break this problem into two subproblems by setting (k + 2.333333334) equal to 2.108185108 and -2.108185108.Subproblem 1
k + 2.333333334 = 2.108185108 Simplifying k + 2.333333334 = 2.108185108 Reorder the terms: 2.333333334 + k = 2.108185108 Solving 2.333333334 + k = 2.108185108 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + k = 2.108185108 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + k = 2.108185108 + -2.333333334 k = 2.108185108 + -2.333333334 Combine like terms: 2.108185108 + -2.333333334 = -0.225148226 k = -0.225148226 Simplifying k = -0.225148226Subproblem 2
k + 2.333333334 = -2.108185108 Simplifying k + 2.333333334 = -2.108185108 Reorder the terms: 2.333333334 + k = -2.108185108 Solving 2.333333334 + k = -2.108185108 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + k = -2.108185108 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + k = -2.108185108 + -2.333333334 k = -2.108185108 + -2.333333334 Combine like terms: -2.108185108 + -2.333333334 = -4.441518442 k = -4.441518442 Simplifying k = -4.441518442Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.225148226, -4.441518442}
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