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Simplifying 0.02P + 0.04P(1800 + -1P) = 80 0.02P + (1800 * 0.04P + -1P * 0.04P) = 80 0.02P + (72P + -0.04P2) = 80 Combine like terms: 0.02P + 72P = 72.02P 72.02P + -0.04P2 = 80 Solving 72.02P + -0.04P2 = 80 Solving for variable 'P'. Reorder the terms: -80 + 72.02P + -0.04P2 = 80 + -80 Combine like terms: 80 + -80 = 0 -80 + 72.02P + -0.04P2 = 0 Begin completing the square. Divide all terms by -0.04 the coefficient of the squared term: Divide each side by '-0.04'. 2000 + -1800.5P + P2 = 0 Move the constant term to the right: Add '-2000' to each side of the equation. 2000 + -1800.5P + -2000 + P2 = 0 + -2000 Reorder the terms: 2000 + -2000 + -1800.5P + P2 = 0 + -2000 Combine like terms: 2000 + -2000 = 0 0 + -1800.5P + P2 = 0 + -2000 -1800.5P + P2 = 0 + -2000 Combine like terms: 0 + -2000 = -2000 -1800.5P + P2 = -2000 The P term is -1800.5P. Take half its coefficient (-900.25). Square it (810450.0625) and add it to both sides. Add '810450.0625' to each side of the equation. -1800.5P + 810450.0625 + P2 = -2000 + 810450.0625 Reorder the terms: 810450.0625 + -1800.5P + P2 = -2000 + 810450.0625 Combine like terms: -2000 + 810450.0625 = 808450.0625 810450.0625 + -1800.5P + P2 = 808450.0625 Factor a perfect square on the left side: (P + -900.25)(P + -900.25) = 808450.0625 Calculate the square root of the right side: 899.138511298 Break this problem into two subproblems by setting (P + -900.25) equal to 899.138511298 and -899.138511298.Subproblem 1
P + -900.25 = 899.138511298 Simplifying P + -900.25 = 899.138511298 Reorder the terms: -900.25 + P = 899.138511298 Solving -900.25 + P = 899.138511298 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '900.25' to each side of the equation. -900.25 + 900.25 + P = 899.138511298 + 900.25 Combine like terms: -900.25 + 900.25 = 0.00 0.00 + P = 899.138511298 + 900.25 P = 899.138511298 + 900.25 Combine like terms: 899.138511298 + 900.25 = 1799.388511298 P = 1799.388511298 Simplifying P = 1799.388511298Subproblem 2
P + -900.25 = -899.138511298 Simplifying P + -900.25 = -899.138511298 Reorder the terms: -900.25 + P = -899.138511298 Solving -900.25 + P = -899.138511298 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '900.25' to each side of the equation. -900.25 + 900.25 + P = -899.138511298 + 900.25 Combine like terms: -900.25 + 900.25 = 0.00 0.00 + P = -899.138511298 + 900.25 P = -899.138511298 + 900.25 Combine like terms: -899.138511298 + 900.25 = 1.111488702 P = 1.111488702 Simplifying P = 1.111488702Solution
The solution to the problem is based on the solutions from the subproblems. P = {1799.388511298, 1.111488702}
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