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Simplifying 0 = x4 + -4x2 + 1 Reorder the terms: 0 = 1 + -4x2 + x4 Solving 0 = 1 + -4x2 + x4 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 4x2 + -1x4 = 1 + -4x2 + x4 + -1 + 4x2 + -1x4 Reorder the terms: -1 + 4x2 + -1x4 = 1 + -1 + -4x2 + 4x2 + x4 + -1x4 Combine like terms: 1 + -1 = 0 -1 + 4x2 + -1x4 = 0 + -4x2 + 4x2 + x4 + -1x4 -1 + 4x2 + -1x4 = -4x2 + 4x2 + x4 + -1x4 Combine like terms: -4x2 + 4x2 = 0 -1 + 4x2 + -1x4 = 0 + x4 + -1x4 -1 + 4x2 + -1x4 = x4 + -1x4 Combine like terms: x4 + -1x4 = 0 -1 + 4x2 + -1x4 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -4x2 + x4 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -4x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4x2 + x4 = 0 + -1 -4x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -4x2 + x4 = -1 The x term is -4x2. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4x2 + 4 + x4 = -1 + 4 Reorder the terms: 4 + -4x2 + x4 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + -4x2 + x4 = 3 Factor a perfect square on the left side: (x2 + -2)(x2 + -2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x2 + -2) equal to 1.732050808 and -1.732050808.Subproblem 1
x2 + -2 = 1.732050808 Simplifying x2 + -2 = 1.732050808 Reorder the terms: -2 + x2 = 1.732050808 Solving -2 + x2 = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = 1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 1.732050808 + 2 x2 = 1.732050808 + 2 Combine like terms: 1.732050808 + 2 = 3.732050808 x2 = 3.732050808 Simplifying x2 = 3.732050808 Take the square root of each side: x = {-1.931851653, 1.931851653}Subproblem 2
x2 + -2 = -1.732050808 Simplifying x2 + -2 = -1.732050808 Reorder the terms: -2 + x2 = -1.732050808 Solving -2 + x2 = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = -1.732050808 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = -1.732050808 + 2 x2 = -1.732050808 + 2 Combine like terms: -1.732050808 + 2 = 0.267949192 x2 = 0.267949192 Simplifying x2 = 0.267949192 Take the square root of each side: x = {-0.51763809, 0.51763809}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.931851653, 1.931851653, -0.51763809, 0.51763809}
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