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Simplifying k4 + -1k2 + -1 = 0 Reorder the terms: -1 + -1k2 + k4 = 0 Solving -1 + -1k2 + k4 = 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 + -1k2 + 1 + k4 = 0 + 1 Reorder the terms: -1 + 1 + -1k2 + k4 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1k2 + k4 = 0 + 1 -1k2 + k4 = 0 + 1 Combine like terms: 0 + 1 = 1 -1k2 + k4 = 1 The k term is -1k2. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1k2 + 0.25 + k4 = 1 + 0.25 Reorder the terms: 0.25 + -1k2 + k4 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1k2 + k4 = 1.25 Factor a perfect square on the left side: (k2 + -0.5)(k2 + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (k2 + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
k2 + -0.5 = 1.118033989 Simplifying k2 + -0.5 = 1.118033989 Reorder the terms: -0.5 + k2 = 1.118033989 Solving -0.5 + k2 = 1.118033989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k2 = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k2 = 1.118033989 + 0.5 k2 = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 k2 = 1.618033989 Simplifying k2 = 1.618033989 Take the square root of each side: k = {-1.27201965, 1.27201965}Subproblem 2
k2 + -0.5 = -1.118033989 Simplifying k2 + -0.5 = -1.118033989 Reorder the terms: -0.5 + k2 = -1.118033989 Solving -0.5 + k2 = -1.118033989 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + k2 = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + k2 = -1.118033989 + 0.5 k2 = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 k2 = -0.618033989 Simplifying k2 = -0.618033989 Reorder the terms: 0.618033989 + k2 = -0.618033989 + 0.618033989 Combine like terms: -0.618033989 + 0.618033989 = 0.000000000 0.618033989 + k2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.
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