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Simplifying 2p2 = -4p + 1 Reorder the terms: 2p2 = 1 + -4p Solving 2p2 = 1 + -4p Solving for variable 'p'. Reorder the terms: -1 + 4p + 2p2 = 1 + -4p + -1 + 4p Reorder the terms: -1 + 4p + 2p2 = 1 + -1 + -4p + 4p Combine like terms: 1 + -1 = 0 -1 + 4p + 2p2 = 0 + -4p + 4p -1 + 4p + 2p2 = -4p + 4p Combine like terms: -4p + 4p = 0 -1 + 4p + 2p2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 2p + p2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 2p + 0.5 + p2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 2p + p2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 2p + p2 = 0 + 0.5 2p + p2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 2p + p2 = 0.5 The p term is 2p. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2p + 1 + p2 = 0.5 + 1 Reorder the terms: 1 + 2p + p2 = 0.5 + 1 Combine like terms: 0.5 + 1 = 1.5 1 + 2p + p2 = 1.5 Factor a perfect square on the left side: (p + 1)(p + 1) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (p + 1) equal to 1.224744871 and -1.224744871.Subproblem 1
p + 1 = 1.224744871 Simplifying p + 1 = 1.224744871 Reorder the terms: 1 + p = 1.224744871 Solving 1 + p = 1.224744871 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + p = 1.224744871 + -1 Combine like terms: 1 + -1 = 0 0 + p = 1.224744871 + -1 p = 1.224744871 + -1 Combine like terms: 1.224744871 + -1 = 0.224744871 p = 0.224744871 Simplifying p = 0.224744871Subproblem 2
p + 1 = -1.224744871 Simplifying p + 1 = -1.224744871 Reorder the terms: 1 + p = -1.224744871 Solving 1 + p = -1.224744871 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + p = -1.224744871 + -1 Combine like terms: 1 + -1 = 0 0 + p = -1.224744871 + -1 p = -1.224744871 + -1 Combine like terms: -1.224744871 + -1 = -2.224744871 p = -2.224744871 Simplifying p = -2.224744871Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.224744871, -2.224744871}
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