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Simplifying 0 = 4x4 + -12x2 + 6 Reorder the terms: 0 = 6 + -12x2 + 4x4 Solving 0 = 6 + -12x2 + 4x4 Solving for variable 'x'. Combine like terms: 0 + -6 = -6 -6 + 12x2 + -4x4 = 6 + -12x2 + 4x4 + -6 + 12x2 + -4x4 Reorder the terms: -6 + 12x2 + -4x4 = 6 + -6 + -12x2 + 12x2 + 4x4 + -4x4 Combine like terms: 6 + -6 = 0 -6 + 12x2 + -4x4 = 0 + -12x2 + 12x2 + 4x4 + -4x4 -6 + 12x2 + -4x4 = -12x2 + 12x2 + 4x4 + -4x4 Combine like terms: -12x2 + 12x2 = 0 -6 + 12x2 + -4x4 = 0 + 4x4 + -4x4 -6 + 12x2 + -4x4 = 4x4 + -4x4 Combine like terms: 4x4 + -4x4 = 0 -6 + 12x2 + -4x4 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 6x2 + -2x4) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 6x2 + -2x4)' equal to zero and attempt to solve: Simplifying -3 + 6x2 + -2x4 = 0 Solving -3 + 6x2 + -2x4 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 1.5 + -3x2 + x4 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3x2 + -1.5 + x4 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3x2 + x4 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3x2 + x4 = 0 + -1.5 -3x2 + x4 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3x2 + x4 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + -3x2 + x4 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3x2 + x4 = 0.75 Factor a perfect square on the left side: (x2 + -1.5)(x2 + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x2 + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x2 + -1.5 = 0.866025404 Simplifying x2 + -1.5 = 0.866025404 Reorder the terms: -1.5 + x2 = 0.866025404 Solving -1.5 + x2 = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = 0.866025404 + 1.5 x2 = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 x2 = 2.366025404 Simplifying x2 = 2.366025404 Take the square root of each side: x = {-1.538189001, 1.538189001}Subproblem 2
x2 + -1.5 = -0.866025404 Simplifying x2 + -1.5 = -0.866025404 Reorder the terms: -1.5 + x2 = -0.866025404 Solving -1.5 + x2 = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x2 = -0.866025404 + 1.5 x2 = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 x2 = 0.633974596 Simplifying x2 = 0.633974596 Take the square root of each side: x = {-0.796225217, 0.796225217}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.538189001, 1.538189001, -0.796225217, 0.796225217}Solution
x = {-1.538189001, 1.538189001, -0.796225217, 0.796225217}
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