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Simplifying (2x + 1)(2x + -1) + -1(2x + 1) * 2 = 8 Reorder the terms: (1 + 2x)(2x + -1) + -1(2x + 1) * 2 = 8 Reorder the terms: (1 + 2x)(-1 + 2x) + -1(2x + 1) * 2 = 8 Multiply (1 + 2x) * (-1 + 2x) (1(-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8 ((-1 * 1 + 2x * 1) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8 ((-1 + 2x) + 2x * (-1 + 2x)) + -1(2x + 1) * 2 = 8 (-1 + 2x + (-1 * 2x + 2x * 2x)) + -1(2x + 1) * 2 = 8 (-1 + 2x + (-2x + 4x2)) + -1(2x + 1) * 2 = 8 Combine like terms: 2x + -2x = 0 (-1 + 0 + 4x2) + -1(2x + 1) * 2 = 8 (-1 + 4x2) + -1(2x + 1) * 2 = 8 Reorder the terms: -1 + 4x2 + -1(1 + 2x) * 2 = 8 Reorder the terms for easier multiplication: -1 + 4x2 + -1 * 2(1 + 2x) = 8 Multiply -1 * 2 -1 + 4x2 + -2(1 + 2x) = 8 -1 + 4x2 + (1 * -2 + 2x * -2) = 8 -1 + 4x2 + (-2 + -4x) = 8 Reorder the terms: -1 + -2 + -4x + 4x2 = 8 Combine like terms: -1 + -2 = -3 -3 + -4x + 4x2 = 8 Solving -3 + -4x + 4x2 = 8 Solving for variable 'x'. Reorder the terms: -3 + -8 + -4x + 4x2 = 8 + -8 Combine like terms: -3 + -8 = -11 -11 + -4x + 4x2 = 8 + -8 Combine like terms: 8 + -8 = 0 -11 + -4x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.75 + -1x + x2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + -1x + 2.75 + x2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + -1x + x2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + -1x + x2 = 0 + 2.75 -1x + x2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 -1x + x2 = 2.75 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 2.75 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 2.75 + 0.25 Combine like terms: 2.75 + 0.25 = 3 0.25 + -1x + x2 = 3 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + -0.5) equal to 1.732050808 and -1.732050808.Subproblem 1
x + -0.5 = 1.732050808 Simplifying x + -0.5 = 1.732050808 Reorder the terms: -0.5 + x = 1.732050808 Solving -0.5 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.732050808 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.732050808 + 0.5 x = 1.732050808 + 0.5 Combine like terms: 1.732050808 + 0.5 = 2.232050808 x = 2.232050808 Simplifying x = 2.232050808Subproblem 2
x + -0.5 = -1.732050808 Simplifying x + -0.5 = -1.732050808 Reorder the terms: -0.5 + x = -1.732050808 Solving -0.5 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.732050808 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.732050808 + 0.5 x = -1.732050808 + 0.5 Combine like terms: -1.732050808 + 0.5 = -1.232050808 x = -1.232050808 Simplifying x = -1.232050808Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.232050808, -1.232050808}
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