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Simplifying 20000 = -0.5x2 + 222x + -160 Reorder the terms: 20000 = -160 + 222x + -0.5x2 Solving 20000 = -160 + 222x + -0.5x2 Solving for variable 'x'. Combine like terms: 20000 + 160 = 20160 20160 + -222x + 0.5x2 = -160 + 222x + -0.5x2 + 160 + -222x + 0.5x2 Reorder the terms: 20160 + -222x + 0.5x2 = -160 + 160 + 222x + -222x + -0.5x2 + 0.5x2 Combine like terms: -160 + 160 = 0 20160 + -222x + 0.5x2 = 0 + 222x + -222x + -0.5x2 + 0.5x2 20160 + -222x + 0.5x2 = 222x + -222x + -0.5x2 + 0.5x2 Combine like terms: 222x + -222x = 0 20160 + -222x + 0.5x2 = 0 + -0.5x2 + 0.5x2 20160 + -222x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 20160 + -222x + 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'. 40320 + -444x + x2 = 0 Move the constant term to the right: Add '-40320' to each side of the equation. 40320 + -444x + -40320 + x2 = 0 + -40320 Reorder the terms: 40320 + -40320 + -444x + x2 = 0 + -40320 Combine like terms: 40320 + -40320 = 0 0 + -444x + x2 = 0 + -40320 -444x + x2 = 0 + -40320 Combine like terms: 0 + -40320 = -40320 -444x + x2 = -40320 The x term is -444x. Take half its coefficient (-222). Square it (49284) and add it to both sides. Add '49284' to each side of the equation. -444x + 49284 + x2 = -40320 + 49284 Reorder the terms: 49284 + -444x + x2 = -40320 + 49284 Combine like terms: -40320 + 49284 = 8964 49284 + -444x + x2 = 8964 Factor a perfect square on the left side: (x + -222)(x + -222) = 8964 Calculate the square root of the right side: 94.678403028 Break this problem into two subproblems by setting (x + -222) equal to 94.678403028 and -94.678403028.Subproblem 1
x + -222 = 94.678403028 Simplifying x + -222 = 94.678403028 Reorder the terms: -222 + x = 94.678403028 Solving -222 + x = 94.678403028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '222' to each side of the equation. -222 + 222 + x = 94.678403028 + 222 Combine like terms: -222 + 222 = 0 0 + x = 94.678403028 + 222 x = 94.678403028 + 222 Combine like terms: 94.678403028 + 222 = 316.678403028 x = 316.678403028 Simplifying x = 316.678403028Subproblem 2
x + -222 = -94.678403028 Simplifying x + -222 = -94.678403028 Reorder the terms: -222 + x = -94.678403028 Solving -222 + x = -94.678403028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '222' to each side of the equation. -222 + 222 + x = -94.678403028 + 222 Combine like terms: -222 + 222 = 0 0 + x = -94.678403028 + 222 x = -94.678403028 + 222 Combine like terms: -94.678403028 + 222 = 127.321596972 x = 127.321596972 Simplifying x = 127.321596972Solution
The solution to the problem is based on the solutions from the subproblems. x = {316.678403028, 127.321596972}
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