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Simplifying 800 = 100x + -0.5x2 + -1(40x + 800) Reorder the terms: 800 = 100x + -0.5x2 + -1(800 + 40x) 800 = 100x + -0.5x2 + (800 * -1 + 40x * -1) 800 = 100x + -0.5x2 + (-800 + -40x) Reorder the terms: 800 = -800 + 100x + -40x + -0.5x2 Combine like terms: 100x + -40x = 60x 800 = -800 + 60x + -0.5x2 Solving 800 = -800 + 60x + -0.5x2 Solving for variable 'x'. Combine like terms: 800 + 800 = 1600 1600 + -60x + 0.5x2 = -800 + 60x + -0.5x2 + 800 + -60x + 0.5x2 Reorder the terms: 1600 + -60x + 0.5x2 = -800 + 800 + 60x + -60x + -0.5x2 + 0.5x2 Combine like terms: -800 + 800 = 0 1600 + -60x + 0.5x2 = 0 + 60x + -60x + -0.5x2 + 0.5x2 1600 + -60x + 0.5x2 = 60x + -60x + -0.5x2 + 0.5x2 Combine like terms: 60x + -60x = 0 1600 + -60x + 0.5x2 = 0 + -0.5x2 + 0.5x2 1600 + -60x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 1600 + -60x + 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'. 3200 + -120x + x2 = 0 Move the constant term to the right: Add '-3200' to each side of the equation. 3200 + -120x + -3200 + x2 = 0 + -3200 Reorder the terms: 3200 + -3200 + -120x + x2 = 0 + -3200 Combine like terms: 3200 + -3200 = 0 0 + -120x + x2 = 0 + -3200 -120x + x2 = 0 + -3200 Combine like terms: 0 + -3200 = -3200 -120x + x2 = -3200 The x term is -120x. Take half its coefficient (-60). Square it (3600) and add it to both sides. Add '3600' to each side of the equation. -120x + 3600 + x2 = -3200 + 3600 Reorder the terms: 3600 + -120x + x2 = -3200 + 3600 Combine like terms: -3200 + 3600 = 400 3600 + -120x + x2 = 400 Factor a perfect square on the left side: (x + -60)(x + -60) = 400 Calculate the square root of the right side: 20 Break this problem into two subproblems by setting (x + -60) equal to 20 and -20.Subproblem 1
x + -60 = 20 Simplifying x + -60 = 20 Reorder the terms: -60 + x = 20 Solving -60 + x = 20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + x = 20 + 60 Combine like terms: -60 + 60 = 0 0 + x = 20 + 60 x = 20 + 60 Combine like terms: 20 + 60 = 80 x = 80 Simplifying x = 80Subproblem 2
x + -60 = -20 Simplifying x + -60 = -20 Reorder the terms: -60 + x = -20 Solving -60 + x = -20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + x = -20 + 60 Combine like terms: -60 + 60 = 0 0 + x = -20 + 60 x = -20 + 60 Combine like terms: -20 + 60 = 40 x = 40 Simplifying x = 40Solution
The solution to the problem is based on the solutions from the subproblems. x = {80, 40}
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