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Simplifying x2 + 1.414213562x = 3 Reorder the terms: 1.414213562x + x2 = 3 Solving 1.414213562x + x2 = 3 Solving for variable 'x'. Reorder the terms: -3 + 1.414213562x + x2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 1.414213562x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.414213562x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + 1.414213562x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.414213562x + x2 = 0 + 3 1.414213562x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.414213562x + x2 = 3 The x term is 1.414213562x. Take half its coefficient (0.707106781). Square it (0.4999999997) and add it to both sides. Add '0.4999999997' to each side of the equation. 1.414213562x + 0.4999999997 + x2 = 3 + 0.4999999997 Reorder the terms: 0.4999999997 + 1.414213562x + x2 = 3 + 0.4999999997 Combine like terms: 3 + 0.4999999997 = 3.4999999997 0.4999999997 + 1.414213562x + x2 = 3.4999999997 Factor a perfect square on the left side: (x + 0.707106781)(x + 0.707106781) = 3.4999999997 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (x + 0.707106781) equal to 1.870828693 and -1.870828693.Subproblem 1
x + 0.707106781 = 1.870828693 Simplifying x + 0.707106781 = 1.870828693 Reorder the terms: 0.707106781 + x = 1.870828693 Solving 0.707106781 + x = 1.870828693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.707106781' to each side of the equation. 0.707106781 + -0.707106781 + x = 1.870828693 + -0.707106781 Combine like terms: 0.707106781 + -0.707106781 = 0.000000000 0.000000000 + x = 1.870828693 + -0.707106781 x = 1.870828693 + -0.707106781 Combine like terms: 1.870828693 + -0.707106781 = 1.163721912 x = 1.163721912 Simplifying x = 1.163721912Subproblem 2
x + 0.707106781 = -1.870828693 Simplifying x + 0.707106781 = -1.870828693 Reorder the terms: 0.707106781 + x = -1.870828693 Solving 0.707106781 + x = -1.870828693 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.707106781' to each side of the equation. 0.707106781 + -0.707106781 + x = -1.870828693 + -0.707106781 Combine like terms: 0.707106781 + -0.707106781 = 0.000000000 0.000000000 + x = -1.870828693 + -0.707106781 x = -1.870828693 + -0.707106781 Combine like terms: -1.870828693 + -0.707106781 = -2.577935474 x = -2.577935474 Simplifying x = -2.577935474Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.163721912, -2.577935474}
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