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Simplifying 20000 = 1500x + -3x2 + -1(20x + 66500) Reorder the terms: 20000 = 1500x + -3x2 + -1(66500 + 20x) 20000 = 1500x + -3x2 + (66500 * -1 + 20x * -1) 20000 = 1500x + -3x2 + (-66500 + -20x) Reorder the terms: 20000 = -66500 + 1500x + -20x + -3x2 Combine like terms: 1500x + -20x = 1480x 20000 = -66500 + 1480x + -3x2 Solving 20000 = -66500 + 1480x + -3x2 Solving for variable 'x'. Combine like terms: 20000 + 66500 = 86500 86500 + -1480x + 3x2 = -66500 + 1480x + -3x2 + 66500 + -1480x + 3x2 Reorder the terms: 86500 + -1480x + 3x2 = -66500 + 66500 + 1480x + -1480x + -3x2 + 3x2 Combine like terms: -66500 + 66500 = 0 86500 + -1480x + 3x2 = 0 + 1480x + -1480x + -3x2 + 3x2 86500 + -1480x + 3x2 = 1480x + -1480x + -3x2 + 3x2 Combine like terms: 1480x + -1480x = 0 86500 + -1480x + 3x2 = 0 + -3x2 + 3x2 86500 + -1480x + 3x2 = -3x2 + 3x2 Combine like terms: -3x2 + 3x2 = 0 86500 + -1480x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 28833.33333 + -493.3333333x + x2 = 0 Move the constant term to the right: Add '-28833.33333' to each side of the equation. 28833.33333 + -493.3333333x + -28833.33333 + x2 = 0 + -28833.33333 Reorder the terms: 28833.33333 + -28833.33333 + -493.3333333x + x2 = 0 + -28833.33333 Combine like terms: 28833.33333 + -28833.33333 = 0.00000 0.00000 + -493.3333333x + x2 = 0 + -28833.33333 -493.3333333x + x2 = 0 + -28833.33333 Combine like terms: 0 + -28833.33333 = -28833.33333 -493.3333333x + x2 = -28833.33333 The x term is -493.3333333x. Take half its coefficient (-246.6666667). Square it (60844.44446) and add it to both sides. Add '60844.44446' to each side of the equation. -493.3333333x + 60844.44446 + x2 = -28833.33333 + 60844.44446 Reorder the terms: 60844.44446 + -493.3333333x + x2 = -28833.33333 + 60844.44446 Combine like terms: -28833.33333 + 60844.44446 = 32011.11113 60844.44446 + -493.3333333x + x2 = 32011.11113 Factor a perfect square on the left side: (x + -246.6666667)(x + -246.6666667) = 32011.11113 Calculate the square root of the right side: 178.916492057 Break this problem into two subproblems by setting (x + -246.6666667) equal to 178.916492057 and -178.916492057.Subproblem 1
x + -246.6666667 = 178.916492057 Simplifying x + -246.6666667 = 178.916492057 Reorder the terms: -246.6666667 + x = 178.916492057 Solving -246.6666667 + x = 178.916492057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '246.6666667' to each side of the equation. -246.6666667 + 246.6666667 + x = 178.916492057 + 246.6666667 Combine like terms: -246.6666667 + 246.6666667 = 0.0000000 0.0000000 + x = 178.916492057 + 246.6666667 x = 178.916492057 + 246.6666667 Combine like terms: 178.916492057 + 246.6666667 = 425.583158757 x = 425.583158757 Simplifying x = 425.583158757Subproblem 2
x + -246.6666667 = -178.916492057 Simplifying x + -246.6666667 = -178.916492057 Reorder the terms: -246.6666667 + x = -178.916492057 Solving -246.6666667 + x = -178.916492057 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '246.6666667' to each side of the equation. -246.6666667 + 246.6666667 + x = -178.916492057 + 246.6666667 Combine like terms: -246.6666667 + 246.6666667 = 0.0000000 0.0000000 + x = -178.916492057 + 246.6666667 x = -178.916492057 + 246.6666667 Combine like terms: -178.916492057 + 246.6666667 = 67.750174643 x = 67.750174643 Simplifying x = 67.750174643Solution
The solution to the problem is based on the solutions from the subproblems. x = {425.583158757, 67.750174643}
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