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Simplifying x(2x + 3) + 1 = 20089 Reorder the terms: x(3 + 2x) + 1 = 20089 (3 * x + 2x * x) + 1 = 20089 (3x + 2x2) + 1 = 20089 Reorder the terms: 1 + 3x + 2x2 = 20089 Solving 1 + 3x + 2x2 = 20089 Solving for variable 'x'. Reorder the terms: 1 + -20089 + 3x + 2x2 = 20089 + -20089 Combine like terms: 1 + -20089 = -20088 -20088 + 3x + 2x2 = 20089 + -20089 Combine like terms: 20089 + -20089 = 0 -20088 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10044 + 1.5x + x2 = 0 Move the constant term to the right: Add '10044' to each side of the equation. -10044 + 1.5x + 10044 + x2 = 0 + 10044 Reorder the terms: -10044 + 10044 + 1.5x + x2 = 0 + 10044 Combine like terms: -10044 + 10044 = 0 0 + 1.5x + x2 = 0 + 10044 1.5x + x2 = 0 + 10044 Combine like terms: 0 + 10044 = 10044 1.5x + x2 = 10044 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 = 10044 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 10044 + 0.5625 Combine like terms: 10044 + 0.5625 = 10044.5625 0.5625 + 1.5x + x2 = 10044.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 10044.5625 Calculate the square root of the right side: 100.222564824 Break this problem into two subproblems by setting (x + 0.75) equal to 100.222564824 and -100.222564824.Subproblem 1
x + 0.75 = 100.222564824 Simplifying x + 0.75 = 100.222564824 Reorder the terms: 0.75 + x = 100.222564824 Solving 0.75 + x = 100.222564824 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 = 100.222564824 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 100.222564824 + -0.75 x = 100.222564824 + -0.75 Combine like terms: 100.222564824 + -0.75 = 99.472564824 x = 99.472564824 Simplifying x = 99.472564824Subproblem 2
x + 0.75 = -100.222564824 Simplifying x + 0.75 = -100.222564824 Reorder the terms: 0.75 + x = -100.222564824 Solving 0.75 + x = -100.222564824 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 = -100.222564824 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -100.222564824 + -0.75 x = -100.222564824 + -0.75 Combine like terms: -100.222564824 + -0.75 = -100.972564824 x = -100.972564824 Simplifying x = -100.972564824Solution
The solution to the problem is based on the solutions from the subproblems. x = {99.472564824, -100.972564824}
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