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Simplifying 2x(2x + 1) = 11 Reorder the terms: 2x(1 + 2x) = 11 (1 * 2x + 2x * 2x) = 11 (2x + 4x2) = 11 Solving 2x + 4x2 = 11 Solving for variable 'x'. Reorder the terms: -11 + 2x + 4x2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 2x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.75 + 0.5x + x2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + 0.5x + 2.75 + x2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + 0.5x + x2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + 0.5x + x2 = 0 + 2.75 0.5x + x2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 0.5x + x2 = 2.75 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 2.75 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 2.75 + 0.0625 Combine like terms: 2.75 + 0.0625 = 2.8125 0.0625 + 0.5x + x2 = 2.8125 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (x + 0.25) equal to 1.677050983 and -1.677050983.Subproblem 1
x + 0.25 = 1.677050983 Simplifying x + 0.25 = 1.677050983 Reorder the terms: 0.25 + x = 1.677050983 Solving 0.25 + x = 1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.677050983 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.677050983 + -0.25 x = 1.677050983 + -0.25 Combine like terms: 1.677050983 + -0.25 = 1.427050983 x = 1.427050983 Simplifying x = 1.427050983Subproblem 2
x + 0.25 = -1.677050983 Simplifying x + 0.25 = -1.677050983 Reorder the terms: 0.25 + x = -1.677050983 Solving 0.25 + x = -1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.677050983 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.677050983 + -0.25 x = -1.677050983 + -0.25 Combine like terms: -1.677050983 + -0.25 = -1.927050983 x = -1.927050983 Simplifying x = -1.927050983Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.427050983, -1.927050983}
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