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Simplifying 5x4 + -10x2 + 1 = 0 Reorder the terms: 1 + -10x2 + 5x4 = 0 Solving 1 + -10x2 + 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 + -2x2 + x4 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -2x2 + -0.2 + x4 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -2x2 + x4 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -2x2 + x4 = 0 + -0.2 -2x2 + x4 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -2x2 + x4 = -0.2 The x term is -2x2. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x2 + 1 + x4 = -0.2 + 1 Reorder the terms: 1 + -2x2 + x4 = -0.2 + 1 Combine like terms: -0.2 + 1 = 0.8 1 + -2x2 + x4 = 0.8 Factor a perfect square on the left side: (x2 + -1)(x2 + -1) = 0.8 Calculate the square root of the right side: 0.894427191 Break this problem into two subproblems by setting (x2 + -1) equal to 0.894427191 and -0.894427191.Subproblem 1
x2 + -1 = 0.894427191 Simplifying x2 + -1 = 0.894427191 Reorder the terms: -1 + x2 = 0.894427191 Solving -1 + x2 = 0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = 0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = 0.894427191 + 1 x2 = 0.894427191 + 1 Combine like terms: 0.894427191 + 1 = 1.894427191 x2 = 1.894427191 Simplifying x2 = 1.894427191 Take the square root of each side: x = {-1.37638192, 1.37638192}Subproblem 2
x2 + -1 = -0.894427191 Simplifying x2 + -1 = -0.894427191 Reorder the terms: -1 + x2 = -0.894427191 Solving -1 + x2 = -0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = -0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = -0.894427191 + 1 x2 = -0.894427191 + 1 Combine like terms: -0.894427191 + 1 = 0.105572809 x2 = 0.105572809 Simplifying x2 = 0.105572809 Take the square root of each side: x = {-0.324919696, 0.324919696}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.37638192, 1.37638192, -0.324919696, 0.324919696}
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