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Simplifying 1.875 = 10(p)(1 + -1p) 1.875 = (1 * 10p + -1p * 10p) 1.875 = (10p + -10p2) Solving 1.875 = 10p + -10p2 Solving for variable 'p'. Reorder the terms: 1.875 + -10p + 10p2 = 10p + -10p + -10p2 + 10p2 Combine like terms: 10p + -10p = 0 1.875 + -10p + 10p2 = 0 + -10p2 + 10p2 1.875 + -10p + 10p2 = -10p2 + 10p2 Combine like terms: -10p2 + 10p2 = 0 1.875 + -10p + 10p2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.1875 + -1p + p2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -1p + -0.1875 + p2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -1p + p2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -1p + p2 = 0 + -0.1875 -1p + p2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -1p + p2 = -0.1875 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 = -0.1875 + 0.25 Reorder the terms: 0.25 + -1p + p2 = -0.1875 + 0.25 Combine like terms: -0.1875 + 0.25 = 0.0625 0.25 + -1p + p2 = 0.0625 Factor a perfect square on the left side: (p + -0.5)(p + -0.5) = 0.0625 Calculate the square root of the right side: 0.25 Break this problem into two subproblems by setting (p + -0.5) equal to 0.25 and -0.25.Subproblem 1
p + -0.5 = 0.25 Simplifying p + -0.5 = 0.25 Reorder the terms: -0.5 + p = 0.25 Solving -0.5 + p = 0.25 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 = 0.25 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = 0.25 + 0.5 p = 0.25 + 0.5 Combine like terms: 0.25 + 0.5 = 0.75 p = 0.75 Simplifying p = 0.75Subproblem 2
p + -0.5 = -0.25 Simplifying p + -0.5 = -0.25 Reorder the terms: -0.5 + p = -0.25 Solving -0.5 + p = -0.25 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 = -0.25 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = -0.25 + 0.5 p = -0.25 + 0.5 Combine like terms: -0.25 + 0.5 = 0.25 p = 0.25 Simplifying p = 0.25Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.75, 0.25}
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