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Simplifying x4 + -32x2 + 1 = 0 Reorder the terms: 1 + -32x2 + x4 = 0 Solving 1 + -32x2 + x4 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -32x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -32x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -32x2 + x4 = 0 + -1 -32x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -32x2 + x4 = -1 The x term is -32x2. Take half its coefficient (-16). Square it (256) and add it to both sides. Add '256' to each side of the equation. -32x2 + 256 + x4 = -1 + 256 Reorder the terms: 256 + -32x2 + x4 = -1 + 256 Combine like terms: -1 + 256 = 255 256 + -32x2 + x4 = 255 Factor a perfect square on the left side: (x2 + -16)(x2 + -16) = 255 Calculate the square root of the right side: 15.968719423 Break this problem into two subproblems by setting (x2 + -16) equal to 15.968719423 and -15.968719423.Subproblem 1
x2 + -16 = 15.968719423 Simplifying x2 + -16 = 15.968719423 Reorder the terms: -16 + x2 = 15.968719423 Solving -16 + x2 = 15.968719423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = 15.968719423 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = 15.968719423 + 16 x2 = 15.968719423 + 16 Combine like terms: 15.968719423 + 16 = 31.968719423 x2 = 31.968719423 Simplifying x2 = 31.968719423 Take the square root of each side: x = {-5.654088735, 5.654088735}Subproblem 2
x2 + -16 = -15.968719423 Simplifying x2 + -16 = -15.968719423 Reorder the terms: -16 + x2 = -15.968719423 Solving -16 + x2 = -15.968719423 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = -15.968719423 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = -15.968719423 + 16 x2 = -15.968719423 + 16 Combine like terms: -15.968719423 + 16 = 0.031280577 x2 = 0.031280577 Simplifying x2 = 0.031280577 Take the square root of each side: x = {-0.176863159, 0.176863159}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-5.654088735, 5.654088735, -0.176863159, 0.176863159}
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