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Simplifying 5x4 + -5x2 + 1 = 0 Reorder the terms: 1 + -5x2 + 5x4 = 0 Solving 1 + -5x2 + 5x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + -1x2 + x4 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -1x2 + -0.2 + x4 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -1x2 + x4 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -1x2 + x4 = 0 + -0.2 -1x2 + x4 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -1x2 + x4 = -0.2 The x term is -1x2. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x2 + 0.25 + x4 = -0.2 + 0.25 Reorder the terms: 0.25 + -1x2 + x4 = -0.2 + 0.25 Combine like terms: -0.2 + 0.25 = 0.05 0.25 + -1x2 + x4 = 0.05 Factor a perfect square on the left side: (x2 + -0.5)(x2 + -0.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (x2 + -0.5) equal to 0.223606798 and -0.223606798.Subproblem 1
x2 + -0.5 = 0.223606798 Simplifying x2 + -0.5 = 0.223606798 Reorder the terms: -0.5 + x2 = 0.223606798 Solving -0.5 + x2 = 0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.223606798 + 0.5 x2 = 0.223606798 + 0.5 Combine like terms: 0.223606798 + 0.5 = 0.723606798 x2 = 0.723606798 Simplifying x2 = 0.723606798 Take the square root of each side: x = {-0.850650808, 0.850650808}Subproblem 2
x2 + -0.5 = -0.223606798 Simplifying x2 + -0.5 = -0.223606798 Reorder the terms: -0.5 + x2 = -0.223606798 Solving -0.5 + x2 = -0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = -0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.223606798 + 0.5 x2 = -0.223606798 + 0.5 Combine like terms: -0.223606798 + 0.5 = 0.276393202 x2 = 0.276393202 Simplifying x2 = 0.276393202 Take the square root of each side: x = {-0.525731112, 0.525731112}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.850650808, 0.850650808, -0.525731112, 0.525731112}
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