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Simplifying (x + -100)(200 + -0.5x) = 0 Reorder the terms: (-100 + x)(200 + -0.5x) = 0 Multiply (-100 + x) * (200 + -0.5x) (-100(200 + -0.5x) + x(200 + -0.5x)) = 0 ((200 * -100 + -0.5x * -100) + x(200 + -0.5x)) = 0 ((-20000 + 50x) + x(200 + -0.5x)) = 0 (-20000 + 50x + (200 * x + -0.5x * x)) = 0 (-20000 + 50x + (200x + -0.5x2)) = 0 Combine like terms: 50x + 200x = 250x (-20000 + 250x + -0.5x2) = 0 Solving -20000 + 250x + -0.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 40000 + -500x + x2 = 0 Move the constant term to the right: Add '-40000' to each side of the equation. 40000 + -500x + -40000 + x2 = 0 + -40000 Reorder the terms: 40000 + -40000 + -500x + x2 = 0 + -40000 Combine like terms: 40000 + -40000 = 0 0 + -500x + x2 = 0 + -40000 -500x + x2 = 0 + -40000 Combine like terms: 0 + -40000 = -40000 -500x + x2 = -40000 The x term is -500x. Take half its coefficient (-250). Square it (62500) and add it to both sides. Add '62500' to each side of the equation. -500x + 62500 + x2 = -40000 + 62500 Reorder the terms: 62500 + -500x + x2 = -40000 + 62500 Combine like terms: -40000 + 62500 = 22500 62500 + -500x + x2 = 22500 Factor a perfect square on the left side: (x + -250)(x + -250) = 22500 Calculate the square root of the right side: 150 Break this problem into two subproblems by setting (x + -250) equal to 150 and -150.Subproblem 1
x + -250 = 150 Simplifying x + -250 = 150 Reorder the terms: -250 + x = 150 Solving -250 + x = 150 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + x = 150 + 250 Combine like terms: -250 + 250 = 0 0 + x = 150 + 250 x = 150 + 250 Combine like terms: 150 + 250 = 400 x = 400 Simplifying x = 400Subproblem 2
x + -250 = -150 Simplifying x + -250 = -150 Reorder the terms: -250 + x = -150 Solving -250 + x = -150 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + x = -150 + 250 Combine like terms: -250 + 250 = 0 0 + x = -150 + 250 x = -150 + 250 Combine like terms: -150 + 250 = 100 x = 100 Simplifying x = 100Solution
The solution to the problem is based on the solutions from the subproblems. x = {400, 100}
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