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Simplifying x4 + 2x3 + -1x2 = 0 Reorder the terms: -1x2 + 2x3 + x4 = 0 Solving -1x2 + 2x3 + x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-1 + 2x + x2) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(-1 + 2x + x2)' equal to zero and attempt to solve: Simplifying -1 + 2x + x2 = 0 Solving -1 + 2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 2x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 2x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2x + x2 = 0 + 1 2x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 2x + x2 = 1 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 1 + 1 Reorder the terms: 1 + 2x + x2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2x + x2 = 2 Factor a perfect square on the left side: (x + 1)(x + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (x + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
x + 1 = 1.414213562 Simplifying x + 1 = 1.414213562 Reorder the terms: 1 + x = 1.414213562 Solving 1 + x = 1.414213562 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 + x = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.414213562 + -1 x = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 x = 0.414213562 Simplifying x = 0.414213562Subproblem 2
x + 1 = -1.414213562 Simplifying x + 1 = -1.414213562 Reorder the terms: 1 + x = -1.414213562 Solving 1 + x = -1.414213562 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 + x = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.414213562 + -1 x = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 x = -2.414213562 Simplifying x = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.414213562, -2.414213562}Solution
x = {0, 0.414213562, -2.414213562}
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