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Simplifying k2 + 8k + 1 = 0 Reorder the terms: 1 + 8k + k2 = 0 Solving 1 + 8k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 8k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + 8k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 8k + k2 = 0 + -1 8k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 8k + k2 = -1 The k term is 8k. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8k + 16 + k2 = -1 + 16 Reorder the terms: 16 + 8k + k2 = -1 + 16 Combine like terms: -1 + 16 = 15 16 + 8k + k2 = 15 Factor a perfect square on the left side: (k + 4)(k + 4) = 15 Calculate the square root of the right side: 3.872983346 Break this problem into two subproblems by setting (k + 4) equal to 3.872983346 and -3.872983346.Subproblem 1
k + 4 = 3.872983346 Simplifying k + 4 = 3.872983346 Reorder the terms: 4 + k = 3.872983346 Solving 4 + k = 3.872983346 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + k = 3.872983346 + -4 Combine like terms: 4 + -4 = 0 0 + k = 3.872983346 + -4 k = 3.872983346 + -4 Combine like terms: 3.872983346 + -4 = -0.127016654 k = -0.127016654 Simplifying k = -0.127016654Subproblem 2
k + 4 = -3.872983346 Simplifying k + 4 = -3.872983346 Reorder the terms: 4 + k = -3.872983346 Solving 4 + k = -3.872983346 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + k = -3.872983346 + -4 Combine like terms: 4 + -4 = 0 0 + k = -3.872983346 + -4 k = -3.872983346 + -4 Combine like terms: -3.872983346 + -4 = -7.872983346 k = -7.872983346 Simplifying k = -7.872983346Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.127016654, -7.872983346}
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