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Simplifying p2 = 5p + 1 Reorder the terms: p2 = 1 + 5p Solving p2 = 1 + 5p Solving for variable 'p'. Reorder the terms: -1 + -5p + p2 = 1 + 5p + -1 + -5p Reorder the terms: -1 + -5p + p2 = 1 + -1 + 5p + -5p Combine like terms: 1 + -1 = 0 -1 + -5p + p2 = 0 + 5p + -5p -1 + -5p + p2 = 5p + -5p Combine like terms: 5p + -5p = 0 -1 + -5p + p2 = 0 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 = 5.192582404 p = 5.192582404 Simplifying p = 5.192582404Subproblem 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 = -0.192582404 p = -0.192582404 Simplifying p = -0.192582404Solution
The solution to the problem is based on the solutions from the subproblems. p = {5.192582404, -0.192582404}
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