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Simplifying 144 = (-1p)(p + -90) Remove parenthesis around (-1p) 144 = -1p(p + -90) Reorder the terms: 144 = -1p(-90 + p) 144 = (-90 * -1p + p * -1p) 144 = (90p + -1p2) Solving 144 = 90p + -1p2 Solving for variable 'p'. Reorder the terms: 144 + -90p + p2 = 90p + -90p + -1p2 + p2 Combine like terms: 90p + -90p = 0 144 + -90p + p2 = 0 + -1p2 + p2 144 + -90p + p2 = -1p2 + p2 Combine like terms: -1p2 + p2 = 0 144 + -90p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '-144' to each side of the equation. 144 + -90p + -144 + p2 = 0 + -144 Reorder the terms: 144 + -144 + -90p + p2 = 0 + -144 Combine like terms: 144 + -144 = 0 0 + -90p + p2 = 0 + -144 -90p + p2 = 0 + -144 Combine like terms: 0 + -144 = -144 -90p + p2 = -144 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 = -144 + 2025 Reorder the terms: 2025 + -90p + p2 = -144 + 2025 Combine like terms: -144 + 2025 = 1881 2025 + -90p + p2 = 1881 Factor a perfect square on the left side: (p + -45)(p + -45) = 1881 Calculate the square root of the right side: 43.370496884 Break this problem into two subproblems by setting (p + -45) equal to 43.370496884 and -43.370496884.Subproblem 1
p + -45 = 43.370496884 Simplifying p + -45 = 43.370496884 Reorder the terms: -45 + p = 43.370496884 Solving -45 + p = 43.370496884 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 = 43.370496884 + 45 Combine like terms: -45 + 45 = 0 0 + p = 43.370496884 + 45 p = 43.370496884 + 45 Combine like terms: 43.370496884 + 45 = 88.370496884 p = 88.370496884 Simplifying p = 88.370496884Subproblem 2
p + -45 = -43.370496884 Simplifying p + -45 = -43.370496884 Reorder the terms: -45 + p = -43.370496884 Solving -45 + p = -43.370496884 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 = -43.370496884 + 45 Combine like terms: -45 + 45 = 0 0 + p = -43.370496884 + 45 p = -43.370496884 + 45 Combine like terms: -43.370496884 + 45 = 1.629503116 p = 1.629503116 Simplifying p = 1.629503116Solution
The solution to the problem is based on the solutions from the subproblems. p = {88.370496884, 1.629503116}
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