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Simplifying x6 + 6x2 + -27x4 = 0 Reorder the terms: 6x2 + -27x4 + x6 = 0 Solving 6x2 + -27x4 + x6 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(6 + -27x2 + x4) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(6 + -27x2 + x4)' equal to zero and attempt to solve: Simplifying 6 + -27x2 + x4 = 0 Solving 6 + -27x2 + x4 = 0 Begin completing the square. Move the constant term to the right: Add '-6' to each side of the equation. 6 + -27x2 + -6 + x4 = 0 + -6 Reorder the terms: 6 + -6 + -27x2 + x4 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -27x2 + x4 = 0 + -6 -27x2 + x4 = 0 + -6 Combine like terms: 0 + -6 = -6 -27x2 + x4 = -6 The x term is -27x2. Take half its coefficient (-13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. -27x2 + 182.25 + x4 = -6 + 182.25 Reorder the terms: 182.25 + -27x2 + x4 = -6 + 182.25 Combine like terms: -6 + 182.25 = 176.25 182.25 + -27x2 + x4 = 176.25 Factor a perfect square on the left side: (x2 + -13.5)(x2 + -13.5) = 176.25 Calculate the square root of the right side: 13.275918047 Break this problem into two subproblems by setting (x2 + -13.5) equal to 13.275918047 and -13.275918047.Subproblem 1
x2 + -13.5 = 13.275918047 Simplifying x2 + -13.5 = 13.275918047 Reorder the terms: -13.5 + x2 = 13.275918047 Solving -13.5 + x2 = 13.275918047 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = 13.275918047 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = 13.275918047 + 13.5 x2 = 13.275918047 + 13.5 Combine like terms: 13.275918047 + 13.5 = 26.775918047 x2 = 26.775918047 Simplifying x2 = 26.775918047 Take the square root of each side: x = {-5.174545202, 5.174545202}Subproblem 2
x2 + -13.5 = -13.275918047 Simplifying x2 + -13.5 = -13.275918047 Reorder the terms: -13.5 + x2 = -13.275918047 Solving -13.5 + x2 = -13.275918047 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x2 = -13.275918047 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x2 = -13.275918047 + 13.5 x2 = -13.275918047 + 13.5 Combine like terms: -13.275918047 + 13.5 = 0.224081953 x2 = 0.224081953 Simplifying x2 = 0.224081953 Take the square root of each side: x = {-0.473372953, 0.473372953}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.174545202, 5.174545202, -0.473372953, 0.473372953}Solution
x = {0, -5.174545202, 5.174545202, -0.473372953, 0.473372953}
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