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Simplifying k2 + 12k + 23 = 0 Reorder the terms: 23 + 12k + k2 = 0 Solving 23 + 12k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + 12k + -23 + k2 = 0 + -23 Reorder the terms: 23 + -23 + 12k + k2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + 12k + k2 = 0 + -23 12k + k2 = 0 + -23 Combine like terms: 0 + -23 = -23 12k + k2 = -23 The k term is 12k. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12k + 36 + k2 = -23 + 36 Reorder the terms: 36 + 12k + k2 = -23 + 36 Combine like terms: -23 + 36 = 13 36 + 12k + k2 = 13 Factor a perfect square on the left side: (k + 6)(k + 6) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (k + 6) equal to 3.605551275 and -3.605551275.Subproblem 1
k + 6 = 3.605551275 Simplifying k + 6 = 3.605551275 Reorder the terms: 6 + k = 3.605551275 Solving 6 + k = 3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + k = 3.605551275 + -6 Combine like terms: 6 + -6 = 0 0 + k = 3.605551275 + -6 k = 3.605551275 + -6 Combine like terms: 3.605551275 + -6 = -2.394448725 k = -2.394448725 Simplifying k = -2.394448725Subproblem 2
k + 6 = -3.605551275 Simplifying k + 6 = -3.605551275 Reorder the terms: 6 + k = -3.605551275 Solving 6 + k = -3.605551275 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + k = -3.605551275 + -6 Combine like terms: 6 + -6 = 0 0 + k = -3.605551275 + -6 k = -3.605551275 + -6 Combine like terms: -3.605551275 + -6 = -9.605551275 k = -9.605551275 Simplifying k = -9.605551275Solution
The solution to the problem is based on the solutions from the subproblems. k = {-2.394448725, -9.605551275}
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