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Simplifying 3p2 + p + -3 = 0 Reorder the terms: -3 + p + 3p2 = 0 Solving -3 + p + 3p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 0.3333333333p + p2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.3333333333p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + 0.3333333333p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.3333333333p + p2 = 0 + 1 0.3333333333p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.3333333333p + p2 = 1 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. 0.3333333333p + 0.25 + p2 = 1 + 0.25 Reorder the terms: 0.25 + 0.3333333333p + p2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + 0.3333333333p + p2 = 1.25 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (p + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
p + 0.5 = 1.118033989 Simplifying p + 0.5 = 1.118033989 Reorder the terms: 0.5 + p = 1.118033989 Solving 0.5 + p = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 1.118033989 + -0.5 p = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 p = 0.618033989 Simplifying p = 0.618033989Subproblem 2
p + 0.5 = -1.118033989 Simplifying p + 0.5 = -1.118033989 Reorder the terms: 0.5 + p = -1.118033989 Solving 0.5 + p = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -1.118033989 + -0.5 p = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 p = -1.618033989 Simplifying p = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.618033989, -1.618033989}
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