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Simplifying 0.05 + 0.01p(22 + -1p) = 0.54 0.05 + (22 * 0.01p + -1p * 0.01p) = 0.54 0.05 + (0.22p + -0.01p2) = 0.54 Solving 0.05 + 0.22p + -0.01p2 = 0.54 Solving for variable 'p'. Reorder the terms: 0.05 + -0.54 + 0.22p + -0.01p2 = 0.54 + -0.54 Combine like terms: 0.05 + -0.54 = -0.49 -0.49 + 0.22p + -0.01p2 = 0.54 + -0.54 Combine like terms: 0.54 + -0.54 = 0.00 -0.49 + 0.22p + -0.01p2 = 0.00 Begin completing the square. Divide all terms by -0.01 the coefficient of the squared term: Divide each side by '-0.01'. 49 + -22p + p2 = 0 Move the constant term to the right: Add '-49' to each side of the equation. 49 + -22p + -49 + p2 = 0 + -49 Reorder the terms: 49 + -49 + -22p + p2 = 0 + -49 Combine like terms: 49 + -49 = 0 0 + -22p + p2 = 0 + -49 -22p + p2 = 0 + -49 Combine like terms: 0 + -49 = -49 -22p + p2 = -49 The p term is -22p. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22p + 121 + p2 = -49 + 121 Reorder the terms: 121 + -22p + p2 = -49 + 121 Combine like terms: -49 + 121 = 72 121 + -22p + p2 = 72 Factor a perfect square on the left side: (p + -11)(p + -11) = 72 Calculate the square root of the right side: 8.485281374 Break this problem into two subproblems by setting (p + -11) equal to 8.485281374 and -8.485281374.Subproblem 1
p + -11 = 8.485281374 Simplifying p + -11 = 8.485281374 Reorder the terms: -11 + p = 8.485281374 Solving -11 + p = 8.485281374 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + p = 8.485281374 + 11 Combine like terms: -11 + 11 = 0 0 + p = 8.485281374 + 11 p = 8.485281374 + 11 Combine like terms: 8.485281374 + 11 = 19.485281374 p = 19.485281374 Simplifying p = 19.485281374Subproblem 2
p + -11 = -8.485281374 Simplifying p + -11 = -8.485281374 Reorder the terms: -11 + p = -8.485281374 Solving -11 + p = -8.485281374 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + p = -8.485281374 + 11 Combine like terms: -11 + 11 = 0 0 + p = -8.485281374 + 11 p = -8.485281374 + 11 Combine like terms: -8.485281374 + 11 = 2.514718626 p = 2.514718626 Simplifying p = 2.514718626Solution
The solution to the problem is based on the solutions from the subproblems. p = {19.485281374, 2.514718626}
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