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Simplifying x4 + -41.4236x2 + 1 = 0 Reorder the terms: 1 + -41.4236x2 + x4 = 0 Solving 1 + -41.4236x2 + 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 + -41.4236x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -41.4236x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -41.4236x2 + x4 = 0 + -1 -41.4236x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -41.4236x2 + x4 = -1 The x term is -41.4236x2. Take half its coefficient (-20.7118). Square it (428.9786592) and add it to both sides. Add '428.9786592' to each side of the equation. -41.4236x2 + 428.9786592 + x4 = -1 + 428.9786592 Reorder the terms: 428.9786592 + -41.4236x2 + x4 = -1 + 428.9786592 Combine like terms: -1 + 428.9786592 = 427.9786592 428.9786592 + -41.4236x2 + x4 = 427.9786592 Factor a perfect square on the left side: (x2 + -20.7118)(x2 + -20.7118) = 427.9786592 Calculate the square root of the right side: 20.687645086 Break this problem into two subproblems by setting (x2 + -20.7118) equal to 20.687645086 and -20.687645086.Subproblem 1
x2 + -20.7118 = 20.687645086 Simplifying x2 + -20.7118 = 20.687645086 Reorder the terms: -20.7118 + x2 = 20.687645086 Solving -20.7118 + x2 = 20.687645086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20.7118' to each side of the equation. -20.7118 + 20.7118 + x2 = 20.687645086 + 20.7118 Combine like terms: -20.7118 + 20.7118 = 0.0000 0.0000 + x2 = 20.687645086 + 20.7118 x2 = 20.687645086 + 20.7118 Combine like terms: 20.687645086 + 20.7118 = 41.399445086 x2 = 41.399445086 Simplifying x2 = 41.399445086 Take the square root of each side: x = {-6.434240055, 6.434240055}Subproblem 2
x2 + -20.7118 = -20.687645086 Simplifying x2 + -20.7118 = -20.687645086 Reorder the terms: -20.7118 + x2 = -20.687645086 Solving -20.7118 + x2 = -20.687645086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20.7118' to each side of the equation. -20.7118 + 20.7118 + x2 = -20.687645086 + 20.7118 Combine like terms: -20.7118 + 20.7118 = 0.0000 0.0000 + x2 = -20.687645086 + 20.7118 x2 = -20.687645086 + 20.7118 Combine like terms: -20.687645086 + 20.7118 = 0.024154914 x2 = 0.024154914 Simplifying x2 = 0.024154914 Take the square root of each side: x = {-0.155418512, 0.155418512}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-6.434240055, 6.434240055, -0.155418512, 0.155418512}
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