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Simplifying 116412 = 505x + -0.5x2 Solving 116412 = 505x + -0.5x2 Solving for variable 'x'. Reorder the terms: 116412 + -505x + 0.5x2 = 505x + -505x + -0.5x2 + 0.5x2 Combine like terms: 505x + -505x = 0 116412 + -505x + 0.5x2 = 0 + -0.5x2 + 0.5x2 116412 + -505x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 116412 + -505x + 0.5x2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 232824 + -1010x + x2 = 0 Move the constant term to the right: Add '-232824' to each side of the equation. 232824 + -1010x + -232824 + x2 = 0 + -232824 Reorder the terms: 232824 + -232824 + -1010x + x2 = 0 + -232824 Combine like terms: 232824 + -232824 = 0 0 + -1010x + x2 = 0 + -232824 -1010x + x2 = 0 + -232824 Combine like terms: 0 + -232824 = -232824 -1010x + x2 = -232824 The x term is -1010x. Take half its coefficient (-505). Square it (255025) and add it to both sides. Add '255025' to each side of the equation. -1010x + 255025 + x2 = -232824 + 255025 Reorder the terms: 255025 + -1010x + x2 = -232824 + 255025 Combine like terms: -232824 + 255025 = 22201 255025 + -1010x + x2 = 22201 Factor a perfect square on the left side: (x + -505)(x + -505) = 22201 Calculate the square root of the right side: 149 Break this problem into two subproblems by setting (x + -505) equal to 149 and -149.Subproblem 1
x + -505 = 149 Simplifying x + -505 = 149 Reorder the terms: -505 + x = 149 Solving -505 + x = 149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '505' to each side of the equation. -505 + 505 + x = 149 + 505 Combine like terms: -505 + 505 = 0 0 + x = 149 + 505 x = 149 + 505 Combine like terms: 149 + 505 = 654 x = 654 Simplifying x = 654Subproblem 2
x + -505 = -149 Simplifying x + -505 = -149 Reorder the terms: -505 + x = -149 Solving -505 + x = -149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '505' to each side of the equation. -505 + 505 + x = -149 + 505 Combine like terms: -505 + 505 = 0 0 + x = -149 + 505 x = -149 + 505 Combine like terms: -149 + 505 = 356 x = 356 Simplifying x = 356Solution
The solution to the problem is based on the solutions from the subproblems. x = {654, 356}
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