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Simplifying -23x2 + 162x + -180 = 0 Reorder the terms: -180 + 162x + -23x2 = 0 Solving -180 + 162x + -23x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -23 the coefficient of the squared term: Divide each side by '-23'. 7.826086957 + -7.043478261x + x2 = 0 Move the constant term to the right: Add '-7.826086957' to each side of the equation. 7.826086957 + -7.043478261x + -7.826086957 + x2 = 0 + -7.826086957 Reorder the terms: 7.826086957 + -7.826086957 + -7.043478261x + x2 = 0 + -7.826086957 Combine like terms: 7.826086957 + -7.826086957 = 0.000000000 0.000000000 + -7.043478261x + x2 = 0 + -7.826086957 -7.043478261x + x2 = 0 + -7.826086957 Combine like terms: 0 + -7.826086957 = -7.826086957 -7.043478261x + x2 = -7.826086957 The x term is -7.043478261x. Take half its coefficient (-3.521739131). Square it (12.40264651) and add it to both sides. Add '12.40264651' to each side of the equation. -7.043478261x + 12.40264651 + x2 = -7.826086957 + 12.40264651 Reorder the terms: 12.40264651 + -7.043478261x + x2 = -7.826086957 + 12.40264651 Combine like terms: -7.826086957 + 12.40264651 = 4.576559553 12.40264651 + -7.043478261x + x2 = 4.576559553 Factor a perfect square on the left side: (x + -3.521739131)(x + -3.521739131) = 4.576559553 Calculate the square root of the right side: 2.139289497 Break this problem into two subproblems by setting (x + -3.521739131) equal to 2.139289497 and -2.139289497.Subproblem 1
x + -3.521739131 = 2.139289497 Simplifying x + -3.521739131 = 2.139289497 Reorder the terms: -3.521739131 + x = 2.139289497 Solving -3.521739131 + x = 2.139289497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.521739131' to each side of the equation. -3.521739131 + 3.521739131 + x = 2.139289497 + 3.521739131 Combine like terms: -3.521739131 + 3.521739131 = 0.000000000 0.000000000 + x = 2.139289497 + 3.521739131 x = 2.139289497 + 3.521739131 Combine like terms: 2.139289497 + 3.521739131 = 5.661028628 x = 5.661028628 Simplifying x = 5.661028628Subproblem 2
x + -3.521739131 = -2.139289497 Simplifying x + -3.521739131 = -2.139289497 Reorder the terms: -3.521739131 + x = -2.139289497 Solving -3.521739131 + x = -2.139289497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.521739131' to each side of the equation. -3.521739131 + 3.521739131 + x = -2.139289497 + 3.521739131 Combine like terms: -3.521739131 + 3.521739131 = 0.000000000 0.000000000 + x = -2.139289497 + 3.521739131 x = -2.139289497 + 3.521739131 Combine like terms: -2.139289497 + 3.521739131 = 1.382449634 x = 1.382449634 Simplifying x = 1.382449634Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.661028628, 1.382449634}
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