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Simplifying 0 = -0.5X2 + 200X + -1000 Reorder the terms: 0 = -1000 + 200X + -0.5X2 Solving 0 = -1000 + 200X + -0.5X2 Solving for variable 'X'. Combine like terms: 0 + 1000 = 1000 1000 + -200X + 0.5X2 = -1000 + 200X + -0.5X2 + 1000 + -200X + 0.5X2 Reorder the terms: 1000 + -200X + 0.5X2 = -1000 + 1000 + 200X + -200X + -0.5X2 + 0.5X2 Combine like terms: -1000 + 1000 = 0 1000 + -200X + 0.5X2 = 0 + 200X + -200X + -0.5X2 + 0.5X2 1000 + -200X + 0.5X2 = 200X + -200X + -0.5X2 + 0.5X2 Combine like terms: 200X + -200X = 0 1000 + -200X + 0.5X2 = 0 + -0.5X2 + 0.5X2 1000 + -200X + 0.5X2 = -0.5X2 + 0.5X2 Combine like terms: -0.5X2 + 0.5X2 = 0.0 1000 + -200X + 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'. 2000 + -400X + X2 = 0 Move the constant term to the right: Add '-2000' to each side of the equation. 2000 + -400X + -2000 + X2 = 0 + -2000 Reorder the terms: 2000 + -2000 + -400X + X2 = 0 + -2000 Combine like terms: 2000 + -2000 = 0 0 + -400X + X2 = 0 + -2000 -400X + X2 = 0 + -2000 Combine like terms: 0 + -2000 = -2000 -400X + X2 = -2000 The X term is -400X. Take half its coefficient (-200). Square it (40000) and add it to both sides. Add '40000' to each side of the equation. -400X + 40000 + X2 = -2000 + 40000 Reorder the terms: 40000 + -400X + X2 = -2000 + 40000 Combine like terms: -2000 + 40000 = 38000 40000 + -400X + X2 = 38000 Factor a perfect square on the left side: (X + -200)(X + -200) = 38000 Calculate the square root of the right side: 194.935886896 Break this problem into two subproblems by setting (X + -200) equal to 194.935886896 and -194.935886896.Subproblem 1
X + -200 = 194.935886896 Simplifying X + -200 = 194.935886896 Reorder the terms: -200 + X = 194.935886896 Solving -200 + X = 194.935886896 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + X = 194.935886896 + 200 Combine like terms: -200 + 200 = 0 0 + X = 194.935886896 + 200 X = 194.935886896 + 200 Combine like terms: 194.935886896 + 200 = 394.935886896 X = 394.935886896 Simplifying X = 394.935886896Subproblem 2
X + -200 = -194.935886896 Simplifying X + -200 = -194.935886896 Reorder the terms: -200 + X = -194.935886896 Solving -200 + X = -194.935886896 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + X = -194.935886896 + 200 Combine like terms: -200 + 200 = 0 0 + X = -194.935886896 + 200 X = -194.935886896 + 200 Combine like terms: -194.935886896 + 200 = 5.064113104 X = 5.064113104 Simplifying X = 5.064113104Solution
The solution to the problem is based on the solutions from the subproblems. X = {394.935886896, 5.064113104}
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