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Simplifying (1 + -1p) * p = 0.2244 Reorder the terms for easier multiplication: p(1 + -1p) = 0.2244 (1 * p + -1p * p) = 0.2244 (1p + -1p2) = 0.2244 Solving 1p + -1p2 = 0.2244 Solving for variable 'p'. Reorder the terms: -0.2244 + 1p + -1p2 = 0.2244 + -0.2244 Combine like terms: 0.2244 + -0.2244 = 0.0000 -0.2244 + 1p + -1p2 = 0.0000 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.2244 + -1p + p2 = 0.0000 Move the constant term to the right: Add '-0.2244' to each side of the equation. 0.2244 + -1p + -0.2244 + p2 = 0.0000 + -0.2244 Reorder the terms: 0.2244 + -0.2244 + -1p + p2 = 0.0000 + -0.2244 Combine like terms: 0.2244 + -0.2244 = 0.0000 0.0000 + -1p + p2 = 0.0000 + -0.2244 -1p + p2 = 0.0000 + -0.2244 Combine like terms: 0.0000 + -0.2244 = -0.2244 -1p + p2 = -0.2244 The p term is -1p. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1p + 0.25 + p2 = -0.2244 + 0.25 Reorder the terms: 0.25 + -1p + p2 = -0.2244 + 0.25 Combine like terms: -0.2244 + 0.25 = 0.0256 0.25 + -1p + p2 = 0.0256 Factor a perfect square on the left side: (p + -0.5)(p + -0.5) = 0.0256 Calculate the square root of the right side: 0.16 Break this problem into two subproblems by setting (p + -0.5) equal to 0.16 and -0.16.Subproblem 1
p + -0.5 = 0.16 Simplifying p + -0.5 = 0.16 Reorder the terms: -0.5 + p = 0.16 Solving -0.5 + p = 0.16 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = 0.16 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = 0.16 + 0.5 p = 0.16 + 0.5 Combine like terms: 0.16 + 0.5 = 0.66 p = 0.66 Simplifying p = 0.66Subproblem 2
p + -0.5 = -0.16 Simplifying p + -0.5 = -0.16 Reorder the terms: -0.5 + p = -0.16 Solving -0.5 + p = -0.16 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = -0.16 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = -0.16 + 0.5 p = -0.16 + 0.5 Combine like terms: -0.16 + 0.5 = 0.34 p = 0.34 Simplifying p = 0.34Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.66, 0.34}
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