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Simplifying (100x + -0.5x2) + -1(60x + 300) = 300 Remove parenthesis around (100x + -0.5x2) 100x + -0.5x2 + -1(60x + 300) = 300 Reorder the terms: 100x + -0.5x2 + -1(300 + 60x) = 300 100x + -0.5x2 + (300 * -1 + 60x * -1) = 300 100x + -0.5x2 + (-300 + -60x) = 300 Reorder the terms: -300 + 100x + -60x + -0.5x2 = 300 Combine like terms: 100x + -60x = 40x -300 + 40x + -0.5x2 = 300 Solving -300 + 40x + -0.5x2 = 300 Solving for variable 'x'. Reorder the terms: -300 + -300 + 40x + -0.5x2 = 300 + -300 Combine like terms: -300 + -300 = -600 -600 + 40x + -0.5x2 = 300 + -300 Combine like terms: 300 + -300 = 0 -600 + 40x + -0.5x2 = 0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 1200 + -80x + x2 = 0 Move the constant term to the right: Add '-1200' to each side of the equation. 1200 + -80x + -1200 + x2 = 0 + -1200 Reorder the terms: 1200 + -1200 + -80x + x2 = 0 + -1200 Combine like terms: 1200 + -1200 = 0 0 + -80x + x2 = 0 + -1200 -80x + x2 = 0 + -1200 Combine like terms: 0 + -1200 = -1200 -80x + x2 = -1200 The x term is -80x. Take half its coefficient (-40). Square it (1600) and add it to both sides. Add '1600' to each side of the equation. -80x + 1600 + x2 = -1200 + 1600 Reorder the terms: 1600 + -80x + x2 = -1200 + 1600 Combine like terms: -1200 + 1600 = 400 1600 + -80x + x2 = 400 Factor a perfect square on the left side: (x + -40)(x + -40) = 400 Calculate the square root of the right side: 20 Break this problem into two subproblems by setting (x + -40) equal to 20 and -20.Subproblem 1
x + -40 = 20 Simplifying x + -40 = 20 Reorder the terms: -40 + x = 20 Solving -40 + x = 20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = 20 + 40 Combine like terms: -40 + 40 = 0 0 + x = 20 + 40 x = 20 + 40 Combine like terms: 20 + 40 = 60 x = 60 Simplifying x = 60Subproblem 2
x + -40 = -20 Simplifying x + -40 = -20 Reorder the terms: -40 + x = -20 Solving -40 + x = -20 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + x = -20 + 40 Combine like terms: -40 + 40 = 0 0 + x = -20 + 40 x = -20 + 40 Combine like terms: -20 + 40 = 20 x = 20 Simplifying x = 20Solution
The solution to the problem is based on the solutions from the subproblems. x = {60, 20}
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