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Simplifying 5p2 + p + 1 = 0 Reorder the terms: 1 + p + 5p2 = 0 Solving 1 + p + 5p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + 0.2p + p2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + 0.2p + -0.2 + p2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + 0.2p + p2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + 0.2p + p2 = 0 + -0.2 0.2p + p2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 0.2p + p2 = -0.2 The p term is p. 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. 0.2p + 0.25 + p2 = -0.2 + 0.25 Reorder the terms: 0.25 + 0.2p + p2 = -0.2 + 0.25 Combine like terms: -0.2 + 0.25 = 0.05 0.25 + 0.2p + p2 = 0.05 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (p + 0.5) equal to 0.223606798 and -0.223606798.Subproblem 1
p + 0.5 = 0.223606798 Simplifying p + 0.5 = 0.223606798 Reorder the terms: 0.5 + p = 0.223606798 Solving 0.5 + p = 0.223606798 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 = 0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 0.223606798 + -0.5 p = 0.223606798 + -0.5 Combine like terms: 0.223606798 + -0.5 = -0.276393202 p = -0.276393202 Simplifying p = -0.276393202Subproblem 2
p + 0.5 = -0.223606798 Simplifying p + 0.5 = -0.223606798 Reorder the terms: 0.5 + p = -0.223606798 Solving 0.5 + p = -0.223606798 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 = -0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -0.223606798 + -0.5 p = -0.223606798 + -0.5 Combine like terms: -0.223606798 + -0.5 = -0.723606798 p = -0.723606798 Simplifying p = -0.723606798Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.276393202, -0.723606798}
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