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Simplifying 1 = -5p2 + 450p + -40 Reorder the terms: 1 = -40 + 450p + -5p2 Solving 1 = -40 + 450p + -5p2 Solving for variable 'p'. Combine like terms: 1 + 40 = 41 41 + -450p + 5p2 = -40 + 450p + -5p2 + 40 + -450p + 5p2 Reorder the terms: 41 + -450p + 5p2 = -40 + 40 + 450p + -450p + -5p2 + 5p2 Combine like terms: -40 + 40 = 0 41 + -450p + 5p2 = 0 + 450p + -450p + -5p2 + 5p2 41 + -450p + 5p2 = 450p + -450p + -5p2 + 5p2 Combine like terms: 450p + -450p = 0 41 + -450p + 5p2 = 0 + -5p2 + 5p2 41 + -450p + 5p2 = -5p2 + 5p2 Combine like terms: -5p2 + 5p2 = 0 41 + -450p + 5p2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 8.2 + -90p + p2 = 0 Move the constant term to the right: Add '-8.2' to each side of the equation. 8.2 + -90p + -8.2 + p2 = 0 + -8.2 Reorder the terms: 8.2 + -8.2 + -90p + p2 = 0 + -8.2 Combine like terms: 8.2 + -8.2 = 0.0 0.0 + -90p + p2 = 0 + -8.2 -90p + p2 = 0 + -8.2 Combine like terms: 0 + -8.2 = -8.2 -90p + p2 = -8.2 The p term is -90p. Take half its coefficient (-45). Square it (2025) and add it to both sides. Add '2025' to each side of the equation. -90p + 2025 + p2 = -8.2 + 2025 Reorder the terms: 2025 + -90p + p2 = -8.2 + 2025 Combine like terms: -8.2 + 2025 = 2016.8 2025 + -90p + p2 = 2016.8 Factor a perfect square on the left side: (p + -45)(p + -45) = 2016.8 Calculate the square root of the right side: 44.908796466 Break this problem into two subproblems by setting (p + -45) equal to 44.908796466 and -44.908796466.Subproblem 1
p + -45 = 44.908796466 Simplifying p + -45 = 44.908796466 Reorder the terms: -45 + p = 44.908796466 Solving -45 + p = 44.908796466 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + p = 44.908796466 + 45 Combine like terms: -45 + 45 = 0 0 + p = 44.908796466 + 45 p = 44.908796466 + 45 Combine like terms: 44.908796466 + 45 = 89.908796466 p = 89.908796466 Simplifying p = 89.908796466Subproblem 2
p + -45 = -44.908796466 Simplifying p + -45 = -44.908796466 Reorder the terms: -45 + p = -44.908796466 Solving -45 + p = -44.908796466 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + p = -44.908796466 + 45 Combine like terms: -45 + 45 = 0 0 + p = -44.908796466 + 45 p = -44.908796466 + 45 Combine like terms: -44.908796466 + 45 = 0.091203534 p = 0.091203534 Simplifying p = 0.091203534Solution
The solution to the problem is based on the solutions from the subproblems. p = {89.908796466, 0.091203534}
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