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Simplifying 6x4 + -81x2 + 162 = 0 Reorder the terms: 162 + -81x2 + 6x4 = 0 Solving 162 + -81x2 + 6x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(54 + -27x2 + 2x4) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(54 + -27x2 + 2x4)' equal to zero and attempt to solve: Simplifying 54 + -27x2 + 2x4 = 0 Solving 54 + -27x2 + 2x4 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 27 + -13.5x2 + x4 = 0 Move the constant term to the right: Add '-27' to each side of the equation. 27 + -13.5x2 + -27 + x4 = 0 + -27 Reorder the terms: 27 + -27 + -13.5x2 + x4 = 0 + -27 Combine like terms: 27 + -27 = 0 0 + -13.5x2 + x4 = 0 + -27 -13.5x2 + x4 = 0 + -27 Combine like terms: 0 + -27 = -27 -13.5x2 + x4 = -27 The x term is -13.5x2. Take half its coefficient (-6.75). Square it (45.5625) and add it to both sides. Add '45.5625' to each side of the equation. -13.5x2 + 45.5625 + x4 = -27 + 45.5625 Reorder the terms: 45.5625 + -13.5x2 + x4 = -27 + 45.5625 Combine like terms: -27 + 45.5625 = 18.5625 45.5625 + -13.5x2 + x4 = 18.5625 Factor a perfect square on the left side: (x2 + -6.75)(x2 + -6.75) = 18.5625 Calculate the square root of the right side: 4.308421985 Break this problem into two subproblems by setting (x2 + -6.75) equal to 4.308421985 and -4.308421985.Subproblem 1
x2 + -6.75 = 4.308421985 Simplifying x2 + -6.75 = 4.308421985 Reorder the terms: -6.75 + x2 = 4.308421985 Solving -6.75 + x2 = 4.308421985 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.75' to each side of the equation. -6.75 + 6.75 + x2 = 4.308421985 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + x2 = 4.308421985 + 6.75 x2 = 4.308421985 + 6.75 Combine like terms: 4.308421985 + 6.75 = 11.058421985 x2 = 11.058421985 Simplifying x2 = 11.058421985 Take the square root of each side: x = {-3.325420573, 3.325420573}Subproblem 2
x2 + -6.75 = -4.308421985 Simplifying x2 + -6.75 = -4.308421985 Reorder the terms: -6.75 + x2 = -4.308421985 Solving -6.75 + x2 = -4.308421985 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.75' to each side of the equation. -6.75 + 6.75 + x2 = -4.308421985 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + x2 = -4.308421985 + 6.75 x2 = -4.308421985 + 6.75 Combine like terms: -4.308421985 + 6.75 = 2.441578015 x2 = 2.441578015 Simplifying x2 = 2.441578015 Take the square root of each side: x = {-1.562554964, 1.562554964}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.325420573, 3.325420573, -1.562554964, 1.562554964}Solution
x = {-3.325420573, 3.325420573, -1.562554964, 1.562554964}
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