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Simplifying x(2x + 3) + 1 = 20088 Reorder the terms: x(3 + 2x) + 1 = 20088 (3 * x + 2x * x) + 1 = 20088 (3x + 2x2) + 1 = 20088 Reorder the terms: 1 + 3x + 2x2 = 20088 Solving 1 + 3x + 2x2 = 20088 Solving for variable 'x'. Reorder the terms: 1 + -20088 + 3x + 2x2 = 20088 + -20088 Combine like terms: 1 + -20088 = -20087 -20087 + 3x + 2x2 = 20088 + -20088 Combine like terms: 20088 + -20088 = 0 -20087 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10043.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '10043.5' to each side of the equation. -10043.5 + 1.5x + 10043.5 + x2 = 0 + 10043.5 Reorder the terms: -10043.5 + 10043.5 + 1.5x + x2 = 0 + 10043.5 Combine like terms: -10043.5 + 10043.5 = 0.0 0.0 + 1.5x + x2 = 0 + 10043.5 1.5x + x2 = 0 + 10043.5 Combine like terms: 0 + 10043.5 = 10043.5 1.5x + x2 = 10043.5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 10043.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 10043.5 + 0.5625 Combine like terms: 10043.5 + 0.5625 = 10044.0625 0.5625 + 1.5x + x2 = 10044.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 10044.0625 Calculate the square root of the right side: 100.220070345 Break this problem into two subproblems by setting (x + 0.75) equal to 100.220070345 and -100.220070345.Subproblem 1
x + 0.75 = 100.220070345 Simplifying x + 0.75 = 100.220070345 Reorder the terms: 0.75 + x = 100.220070345 Solving 0.75 + x = 100.220070345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 100.220070345 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 100.220070345 + -0.75 x = 100.220070345 + -0.75 Combine like terms: 100.220070345 + -0.75 = 99.470070345 x = 99.470070345 Simplifying x = 99.470070345Subproblem 2
x + 0.75 = -100.220070345 Simplifying x + 0.75 = -100.220070345 Reorder the terms: 0.75 + x = -100.220070345 Solving 0.75 + x = -100.220070345 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -100.220070345 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -100.220070345 + -0.75 x = -100.220070345 + -0.75 Combine like terms: -100.220070345 + -0.75 = -100.970070345 x = -100.970070345 Simplifying x = -100.970070345Solution
The solution to the problem is based on the solutions from the subproblems. x = {99.470070345, -100.970070345}
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