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Simplifying 16p2 + 8p = 19 Reorder the terms: 8p + 16p2 = 19 Solving 8p + 16p2 = 19 Solving for variable 'p'. Reorder the terms: -19 + 8p + 16p2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 8p + 16p2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.1875 + 0.5p + p2 = 0 Move the constant term to the right: Add '1.1875' to each side of the equation. -1.1875 + 0.5p + 1.1875 + p2 = 0 + 1.1875 Reorder the terms: -1.1875 + 1.1875 + 0.5p + p2 = 0 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + 0.5p + p2 = 0 + 1.1875 0.5p + p2 = 0 + 1.1875 Combine like terms: 0 + 1.1875 = 1.1875 0.5p + p2 = 1.1875 The p term is 0.5p. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5p + 0.0625 + p2 = 1.1875 + 0.0625 Reorder the terms: 0.0625 + 0.5p + p2 = 1.1875 + 0.0625 Combine like terms: 1.1875 + 0.0625 = 1.25 0.0625 + 0.5p + p2 = 1.25 Factor a perfect square on the left side: (p + 0.25)(p + 0.25) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (p + 0.25) equal to 1.118033989 and -1.118033989.Subproblem 1
p + 0.25 = 1.118033989 Simplifying p + 0.25 = 1.118033989 Reorder the terms: 0.25 + p = 1.118033989 Solving 0.25 + p = 1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + p = 1.118033989 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + p = 1.118033989 + -0.25 p = 1.118033989 + -0.25 Combine like terms: 1.118033989 + -0.25 = 0.868033989 p = 0.868033989 Simplifying p = 0.868033989Subproblem 2
p + 0.25 = -1.118033989 Simplifying p + 0.25 = -1.118033989 Reorder the terms: 0.25 + p = -1.118033989 Solving 0.25 + p = -1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + p = -1.118033989 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + p = -1.118033989 + -0.25 p = -1.118033989 + -0.25 Combine like terms: -1.118033989 + -0.25 = -1.368033989 p = -1.368033989 Simplifying p = -1.368033989Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.868033989, -1.368033989}
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