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Simplifying 3x4 + -16x2 + 3 = 0 Reorder the terms: 3 + -16x2 + 3x4 = 0 Solving 3 + -16x2 + 3x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1 + -5.333333333x2 + x4 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -5.333333333x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -5.333333333x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -5.333333333x2 + x4 = 0 + -1 -5.333333333x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -5.333333333x2 + x4 = -1 The x term is -5.333333333x2. Take half its coefficient (-2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. -5.333333333x2 + 7.111111113 + x4 = -1 + 7.111111113 Reorder the terms: 7.111111113 + -5.333333333x2 + x4 = -1 + 7.111111113 Combine like terms: -1 + 7.111111113 = 6.111111113 7.111111113 + -5.333333333x2 + x4 = 6.111111113 Factor a perfect square on the left side: (x2 + -2.666666667)(x2 + -2.666666667) = 6.111111113 Calculate the square root of the right side: 2.472066163 Break this problem into two subproblems by setting (x2 + -2.666666667) equal to 2.472066163 and -2.472066163.Subproblem 1
x2 + -2.666666667 = 2.472066163 Simplifying x2 + -2.666666667 = 2.472066163 Reorder the terms: -2.666666667 + x2 = 2.472066163 Solving -2.666666667 + x2 = 2.472066163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x2 = 2.472066163 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x2 = 2.472066163 + 2.666666667 x2 = 2.472066163 + 2.666666667 Combine like terms: 2.472066163 + 2.666666667 = 5.13873283 x2 = 5.13873283 Simplifying x2 = 5.13873283 Take the square root of each side: x = {-2.26687733, 2.26687733}Subproblem 2
x2 + -2.666666667 = -2.472066163 Simplifying x2 + -2.666666667 = -2.472066163 Reorder the terms: -2.666666667 + x2 = -2.472066163 Solving -2.666666667 + x2 = -2.472066163 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x2 = -2.472066163 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x2 = -2.472066163 + 2.666666667 x2 = -2.472066163 + 2.666666667 Combine like terms: -2.472066163 + 2.666666667 = 0.194600504 x2 = 0.194600504 Simplifying x2 = 0.194600504 Take the square root of each side: x = {-0.441135471, 0.441135471}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.26687733, 2.26687733, -0.441135471, 0.441135471}
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