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Simplifying 2x(2x + 1) = 105 Reorder the terms: 2x(1 + 2x) = 105 (1 * 2x + 2x * 2x) = 105 (2x + 4x2) = 105 Solving 2x + 4x2 = 105 Solving for variable 'x'. Reorder the terms: -105 + 2x + 4x2 = 105 + -105 Combine like terms: 105 + -105 = 0 -105 + 2x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -26.25 + 0.5x + x2 = 0 Move the constant term to the right: Add '26.25' to each side of the equation. -26.25 + 0.5x + 26.25 + x2 = 0 + 26.25 Reorder the terms: -26.25 + 26.25 + 0.5x + x2 = 0 + 26.25 Combine like terms: -26.25 + 26.25 = 0.00 0.00 + 0.5x + x2 = 0 + 26.25 0.5x + x2 = 0 + 26.25 Combine like terms: 0 + 26.25 = 26.25 0.5x + x2 = 26.25 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 = 26.25 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 26.25 + 0.0625 Combine like terms: 26.25 + 0.0625 = 26.3125 0.0625 + 0.5x + x2 = 26.3125 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 26.3125 Calculate the square root of the right side: 5.129571132 Break this problem into two subproblems by setting (x + 0.25) equal to 5.129571132 and -5.129571132.Subproblem 1
x + 0.25 = 5.129571132 Simplifying x + 0.25 = 5.129571132 Reorder the terms: 0.25 + x = 5.129571132 Solving 0.25 + x = 5.129571132 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 = 5.129571132 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.129571132 + -0.25 x = 5.129571132 + -0.25 Combine like terms: 5.129571132 + -0.25 = 4.879571132 x = 4.879571132 Simplifying x = 4.879571132Subproblem 2
x + 0.25 = -5.129571132 Simplifying x + 0.25 = -5.129571132 Reorder the terms: 0.25 + x = -5.129571132 Solving 0.25 + x = -5.129571132 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 = -5.129571132 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.129571132 + -0.25 x = -5.129571132 + -0.25 Combine like terms: -5.129571132 + -0.25 = -5.379571132 x = -5.379571132 Simplifying x = -5.379571132Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.879571132, -5.379571132}
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