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Simplifying 11500 = 0.5x2 + 15x + 5000 Reorder the terms: 11500 = 5000 + 15x + 0.5x2 Solving 11500 = 5000 + 15x + 0.5x2 Solving for variable 'x'. Combine like terms: 11500 + -5000 = 6500 6500 + -15x + -0.5x2 = 5000 + 15x + 0.5x2 + -5000 + -15x + -0.5x2 Reorder the terms: 6500 + -15x + -0.5x2 = 5000 + -5000 + 15x + -15x + 0.5x2 + -0.5x2 Combine like terms: 5000 + -5000 = 0 6500 + -15x + -0.5x2 = 0 + 15x + -15x + 0.5x2 + -0.5x2 6500 + -15x + -0.5x2 = 15x + -15x + 0.5x2 + -0.5x2 Combine like terms: 15x + -15x = 0 6500 + -15x + -0.5x2 = 0 + 0.5x2 + -0.5x2 6500 + -15x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 6500 + -15x + -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'. -13000 + 30x + x2 = 0 Move the constant term to the right: Add '13000' to each side of the equation. -13000 + 30x + 13000 + x2 = 0 + 13000 Reorder the terms: -13000 + 13000 + 30x + x2 = 0 + 13000 Combine like terms: -13000 + 13000 = 0 0 + 30x + x2 = 0 + 13000 30x + x2 = 0 + 13000 Combine like terms: 0 + 13000 = 13000 30x + x2 = 13000 The x term is 30x. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30x + 225 + x2 = 13000 + 225 Reorder the terms: 225 + 30x + x2 = 13000 + 225 Combine like terms: 13000 + 225 = 13225 225 + 30x + x2 = 13225 Factor a perfect square on the left side: (x + 15)(x + 15) = 13225 Calculate the square root of the right side: 115 Break this problem into two subproblems by setting (x + 15) equal to 115 and -115.Subproblem 1
x + 15 = 115 Simplifying x + 15 = 115 Reorder the terms: 15 + x = 115 Solving 15 + x = 115 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + x = 115 + -15 Combine like terms: 15 + -15 = 0 0 + x = 115 + -15 x = 115 + -15 Combine like terms: 115 + -15 = 100 x = 100 Simplifying x = 100Subproblem 2
x + 15 = -115 Simplifying x + 15 = -115 Reorder the terms: 15 + x = -115 Solving 15 + x = -115 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + x = -115 + -15 Combine like terms: 15 + -15 = 0 0 + x = -115 + -15 x = -115 + -15 Combine like terms: -115 + -15 = -130 x = -130 Simplifying x = -130Solution
The solution to the problem is based on the solutions from the subproblems. x = {100, -130}
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