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Simplifying 92 = (2x + 1)(x) Reorder the terms: 92 = (1 + 2x)(x) Reorder the terms for easier multiplication: 92 = x(1 + 2x) 92 = (1 * x + 2x * x) 92 = (1x + 2x2) Solving 92 = 1x + 2x2 Solving for variable 'x'. Reorder the terms: 92 + -1x + -2x2 = 1x + -1x + 2x2 + -2x2 Combine like terms: 1x + -1x = 0 92 + -1x + -2x2 = 0 + 2x2 + -2x2 92 + -1x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 92 + -1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -46 + 0.5x + x2 = 0 Move the constant term to the right: Add '46' to each side of the equation. -46 + 0.5x + 46 + x2 = 0 + 46 Reorder the terms: -46 + 46 + 0.5x + x2 = 0 + 46 Combine like terms: -46 + 46 = 0 0 + 0.5x + x2 = 0 + 46 0.5x + x2 = 0 + 46 Combine like terms: 0 + 46 = 46 0.5x + x2 = 46 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 = 46 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 46 + 0.0625 Combine like terms: 46 + 0.0625 = 46.0625 0.0625 + 0.5x + x2 = 46.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 46.0625 Calculate the square root of the right side: 6.78693598 Break this problem into two subproblems by setting (x + 0.25) equal to 6.78693598 and -6.78693598.Subproblem 1
x + 0.25 = 6.78693598 Simplifying x + 0.25 = 6.78693598 Reorder the terms: 0.25 + x = 6.78693598 Solving 0.25 + x = 6.78693598 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 = 6.78693598 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 6.78693598 + -0.25 x = 6.78693598 + -0.25 Combine like terms: 6.78693598 + -0.25 = 6.53693598 x = 6.53693598 Simplifying x = 6.53693598Subproblem 2
x + 0.25 = -6.78693598 Simplifying x + 0.25 = -6.78693598 Reorder the terms: 0.25 + x = -6.78693598 Solving 0.25 + x = -6.78693598 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 = -6.78693598 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -6.78693598 + -0.25 x = -6.78693598 + -0.25 Combine like terms: -6.78693598 + -0.25 = -7.03693598 x = -7.03693598 Simplifying x = -7.03693598Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.53693598, -7.03693598}
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