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Simplifying x4 + -23x2 + 1 = 0 Reorder the terms: 1 + -23x2 + x4 = 0 Solving 1 + -23x2 + 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 + -23x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -23x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -23x2 + x4 = 0 + -1 -23x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -23x2 + x4 = -1 The x term is -23x2. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23x2 + 132.25 + x4 = -1 + 132.25 Reorder the terms: 132.25 + -23x2 + x4 = -1 + 132.25 Combine like terms: -1 + 132.25 = 131.25 132.25 + -23x2 + x4 = 131.25 Factor a perfect square on the left side: (x2 + -11.5)(x2 + -11.5) = 131.25 Calculate the square root of the right side: 11.456439237 Break this problem into two subproblems by setting (x2 + -11.5) equal to 11.456439237 and -11.456439237.Subproblem 1
x2 + -11.5 = 11.456439237 Simplifying x2 + -11.5 = 11.456439237 Reorder the terms: -11.5 + x2 = 11.456439237 Solving -11.5 + x2 = 11.456439237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x2 = 11.456439237 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x2 = 11.456439237 + 11.5 x2 = 11.456439237 + 11.5 Combine like terms: 11.456439237 + 11.5 = 22.956439237 x2 = 22.956439237 Simplifying x2 = 22.956439237 Take the square root of each side: x = {-4.791287847, 4.791287847}Subproblem 2
x2 + -11.5 = -11.456439237 Simplifying x2 + -11.5 = -11.456439237 Reorder the terms: -11.5 + x2 = -11.456439237 Solving -11.5 + x2 = -11.456439237 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x2 = -11.456439237 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x2 = -11.456439237 + 11.5 x2 = -11.456439237 + 11.5 Combine like terms: -11.456439237 + 11.5 = 0.043560763 x2 = 0.043560763 Simplifying x2 = 0.043560763 Take the square root of each side: x = {-0.208712153, 0.208712153}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-4.791287847, 4.791287847, -0.208712153, 0.208712153}
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