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Simplifying 11500 = 0.05x2 + 15x + 6500 Reorder the terms: 11500 = 6500 + 15x + 0.05x2 Solving 11500 = 6500 + 15x + 0.05x2 Solving for variable 'x'. Combine like terms: 11500 + -6500 = 5000 5000 + -15x + -0.05x2 = 6500 + 15x + 0.05x2 + -6500 + -15x + -0.05x2 Reorder the terms: 5000 + -15x + -0.05x2 = 6500 + -6500 + 15x + -15x + 0.05x2 + -0.05x2 Combine like terms: 6500 + -6500 = 0 5000 + -15x + -0.05x2 = 0 + 15x + -15x + 0.05x2 + -0.05x2 5000 + -15x + -0.05x2 = 15x + -15x + 0.05x2 + -0.05x2 Combine like terms: 15x + -15x = 0 5000 + -15x + -0.05x2 = 0 + 0.05x2 + -0.05x2 5000 + -15x + -0.05x2 = 0.05x2 + -0.05x2 Combine like terms: 0.05x2 + -0.05x2 = 0.00 5000 + -15x + -0.05x2 = 0.00 Begin completing the square. Divide all terms by -0.05 the coefficient of the squared term: Divide each side by '-0.05'. -100000 + 300x + x2 = 0 Move the constant term to the right: Add '100000' to each side of the equation. -100000 + 300x + 100000 + x2 = 0 + 100000 Reorder the terms: -100000 + 100000 + 300x + x2 = 0 + 100000 Combine like terms: -100000 + 100000 = 0 0 + 300x + x2 = 0 + 100000 300x + x2 = 0 + 100000 Combine like terms: 0 + 100000 = 100000 300x + x2 = 100000 The x term is 300x. Take half its coefficient (150). Square it (22500) and add it to both sides. Add '22500' to each side of the equation. 300x + 22500 + x2 = 100000 + 22500 Reorder the terms: 22500 + 300x + x2 = 100000 + 22500 Combine like terms: 100000 + 22500 = 122500 22500 + 300x + x2 = 122500 Factor a perfect square on the left side: (x + 150)(x + 150) = 122500 Calculate the square root of the right side: 350 Break this problem into two subproblems by setting (x + 150) equal to 350 and -350.Subproblem 1
x + 150 = 350 Simplifying x + 150 = 350 Reorder the terms: 150 + x = 350 Solving 150 + x = 350 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-150' to each side of the equation. 150 + -150 + x = 350 + -150 Combine like terms: 150 + -150 = 0 0 + x = 350 + -150 x = 350 + -150 Combine like terms: 350 + -150 = 200 x = 200 Simplifying x = 200Subproblem 2
x + 150 = -350 Simplifying x + 150 = -350 Reorder the terms: 150 + x = -350 Solving 150 + x = -350 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-150' to each side of the equation. 150 + -150 + x = -350 + -150 Combine like terms: 150 + -150 = 0 0 + x = -350 + -150 x = -350 + -150 Combine like terms: -350 + -150 = -500 x = -500 Simplifying x = -500Solution
The solution to the problem is based on the solutions from the subproblems. x = {200, -500}
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