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Simplifying 15 = 0.004(x2 + 5x + 1500) Reorder the terms: 15 = 0.004(1500 + 5x + x2) 15 = (1500 * 0.004 + 5x * 0.004 + x2 * 0.004) 15 = (6 + 0.02x + 0.004x2) Solving 15 = 6 + 0.02x + 0.004x2 Solving for variable 'x'. Combine like terms: 15 + -6 = 9 9 + -0.02x + -0.004x2 = 6 + 0.02x + 0.004x2 + -6 + -0.02x + -0.004x2 Reorder the terms: 9 + -0.02x + -0.004x2 = 6 + -6 + 0.02x + -0.02x + 0.004x2 + -0.004x2 Combine like terms: 6 + -6 = 0 9 + -0.02x + -0.004x2 = 0 + 0.02x + -0.02x + 0.004x2 + -0.004x2 9 + -0.02x + -0.004x2 = 0.02x + -0.02x + 0.004x2 + -0.004x2 Combine like terms: 0.02x + -0.02x = 0.00 9 + -0.02x + -0.004x2 = 0.00 + 0.004x2 + -0.004x2 9 + -0.02x + -0.004x2 = 0.004x2 + -0.004x2 Combine like terms: 0.004x2 + -0.004x2 = 0.000 9 + -0.02x + -0.004x2 = 0.000 Begin completing the square. Divide all terms by -0.004 the coefficient of the squared term: Divide each side by '-0.004'. -2250 + 5x + x2 = 0 Move the constant term to the right: Add '2250' to each side of the equation. -2250 + 5x + 2250 + x2 = 0 + 2250 Reorder the terms: -2250 + 2250 + 5x + x2 = 0 + 2250 Combine like terms: -2250 + 2250 = 0 0 + 5x + x2 = 0 + 2250 5x + x2 = 0 + 2250 Combine like terms: 0 + 2250 = 2250 5x + x2 = 2250 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 2250 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 2250 + 6.25 Combine like terms: 2250 + 6.25 = 2256.25 6.25 + 5x + x2 = 2256.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 2256.25 Calculate the square root of the right side: 47.5 Break this problem into two subproblems by setting (x + 2.5) equal to 47.5 and -47.5.Subproblem 1
x + 2.5 = 47.5 Simplifying x + 2.5 = 47.5 Reorder the terms: 2.5 + x = 47.5 Solving 2.5 + x = 47.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 47.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 47.5 + -2.5 x = 47.5 + -2.5 Combine like terms: 47.5 + -2.5 = 45 x = 45 Simplifying x = 45Subproblem 2
x + 2.5 = -47.5 Simplifying x + 2.5 = -47.5 Reorder the terms: 2.5 + x = -47.5 Solving 2.5 + x = -47.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -47.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -47.5 + -2.5 x = -47.5 + -2.5 Combine like terms: -47.5 + -2.5 = -50 x = -50 Simplifying x = -50Solution
The solution to the problem is based on the solutions from the subproblems. x = {45, -50}
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