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Simplifying 1.666 = x2 + 2x Reorder the terms: 1.666 = 2x + x2 Solving 1.666 = 2x + x2 Solving for variable 'x'. Reorder the terms: 1.666 + -2x + -1x2 = 2x + -2x + x2 + -1x2 Combine like terms: 2x + -2x = 0 1.666 + -2x + -1x2 = 0 + x2 + -1x2 1.666 + -2x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 1.666 + -2x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1.666 + 2x + x2 = 0 Move the constant term to the right: Add '1.666' to each side of the equation. -1.666 + 2x + 1.666 + x2 = 0 + 1.666 Reorder the terms: -1.666 + 1.666 + 2x + x2 = 0 + 1.666 Combine like terms: -1.666 + 1.666 = 0.000 0.000 + 2x + x2 = 0 + 1.666 2x + x2 = 0 + 1.666 Combine like terms: 0 + 1.666 = 1.666 2x + x2 = 1.666 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 = 1.666 + 1 Reorder the terms: 1 + 2x + x2 = 1.666 + 1 Combine like terms: 1.666 + 1 = 2.666 1 + 2x + x2 = 2.666 Factor a perfect square on the left side: (x + 1)(x + 1) = 2.666 Calculate the square root of the right side: 1.632789025 Break this problem into two subproblems by setting (x + 1) equal to 1.632789025 and -1.632789025.Subproblem 1
x + 1 = 1.632789025 Simplifying x + 1 = 1.632789025 Reorder the terms: 1 + x = 1.632789025 Solving 1 + x = 1.632789025 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 = 1.632789025 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.632789025 + -1 x = 1.632789025 + -1 Combine like terms: 1.632789025 + -1 = 0.632789025 x = 0.632789025 Simplifying x = 0.632789025Subproblem 2
x + 1 = -1.632789025 Simplifying x + 1 = -1.632789025 Reorder the terms: 1 + x = -1.632789025 Solving 1 + x = -1.632789025 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 = -1.632789025 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.632789025 + -1 x = -1.632789025 + -1 Combine like terms: -1.632789025 + -1 = -2.632789025 x = -2.632789025 Simplifying x = -2.632789025Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.632789025, -2.632789025}
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