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Simplifying p2 + 5p + -1 = 0 Reorder the terms: -1 + 5p + p2 = 0 Solving -1 + 5p + p2 = 0 Solving for variable 'p'. Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 5p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + 5p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 5p + p2 = 0 + 1 5p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 5p + p2 = 1 The p term is 5p. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5p + 6.25 + p2 = 1 + 6.25 Reorder the terms: 6.25 + 5p + p2 = 1 + 6.25 Combine like terms: 1 + 6.25 = 7.25 6.25 + 5p + p2 = 7.25 Factor a perfect square on the left side: (p + 2.5)(p + 2.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (p + 2.5) equal to 2.692582404 and -2.692582404.Subproblem 1
p + 2.5 = 2.692582404 Simplifying p + 2.5 = 2.692582404 Reorder the terms: 2.5 + p = 2.692582404 Solving 2.5 + p = 2.692582404 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + p = 2.692582404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + p = 2.692582404 + -2.5 p = 2.692582404 + -2.5 Combine like terms: 2.692582404 + -2.5 = 0.192582404 p = 0.192582404 Simplifying p = 0.192582404Subproblem 2
p + 2.5 = -2.692582404 Simplifying p + 2.5 = -2.692582404 Reorder the terms: 2.5 + p = -2.692582404 Solving 2.5 + p = -2.692582404 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + p = -2.692582404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + p = -2.692582404 + -2.5 p = -2.692582404 + -2.5 Combine like terms: -2.692582404 + -2.5 = -5.192582404 p = -5.192582404 Simplifying p = -5.192582404Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.192582404, -5.192582404}
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