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Simplifying x5 + -3x3 + x = 0 Reorder the terms: x + -3x3 + x5 = 0 Solving x + -3x3 + x5 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(1 + -3x2 + x4) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(1 + -3x2 + x4)' equal to zero and attempt to solve: Simplifying 1 + -3x2 + x4 = 0 Solving 1 + -3x2 + x4 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -3x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3x2 + x4 = 0 + -1 -3x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -3x2 + x4 = -1 The x term is -3x2. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x2 + 2.25 + x4 = -1 + 2.25 Reorder the terms: 2.25 + -3x2 + x4 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3x2 + x4 = 1.25 Factor a perfect square on the left side: (x2 + -1.5)(x2 + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x2 + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x2 + -1.5 = 1.118033989 Simplifying x2 + -1.5 = 1.118033989 Reorder the terms: -1.5 + x2 = 1.118033989 Solving -1.5 + x2 = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = 1.118033989 + 1.5 x2 = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 x2 = 2.618033989 Simplifying x2 = 2.618033989 Take the square root of each side: x = {-1.618033989, 1.618033989}Subproblem 2
x2 + -1.5 = -1.118033989 Simplifying x2 + -1.5 = -1.118033989 Reorder the terms: -1.5 + x2 = -1.118033989 Solving -1.5 + x2 = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x2 = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = -1.118033989 + 1.5 x2 = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 x2 = 0.381966011 Simplifying x2 = 0.381966011 Take the square root of each side: x = {-0.618033989, 0.618033989}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.618033989, 1.618033989, -0.618033989, 0.618033989}Solution
x = {0, -1.618033989, 1.618033989, -0.618033989, 0.618033989}
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