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Simplifying 3p2 + -6p + 1 = 0 Reorder the terms: 1 + -6p + 3p2 = 0 Solving 1 + -6p + 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'. 0.3333333333 + -2p + p2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -2p + -0.3333333333 + p2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -2p + p2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -2p + p2 = 0 + -0.3333333333 -2p + p2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -2p + p2 = -0.3333333333 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 = -0.3333333333 + 1 Reorder the terms: 1 + -2p + p2 = -0.3333333333 + 1 Combine like terms: -0.3333333333 + 1 = 0.6666666667 1 + -2p + p2 = 0.6666666667 Factor a perfect square on the left side: (p + -1)(p + -1) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (p + -1) equal to 0.816496581 and -0.816496581.Subproblem 1
p + -1 = 0.816496581 Simplifying p + -1 = 0.816496581 Reorder the terms: -1 + p = 0.816496581 Solving -1 + p = 0.816496581 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 = 0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.816496581 + 1 p = 0.816496581 + 1 Combine like terms: 0.816496581 + 1 = 1.816496581 p = 1.816496581 Simplifying p = 1.816496581Subproblem 2
p + -1 = -0.816496581 Simplifying p + -1 = -0.816496581 Reorder the terms: -1 + p = -0.816496581 Solving -1 + p = -0.816496581 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 = -0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.816496581 + 1 p = -0.816496581 + 1 Combine like terms: -0.816496581 + 1 = 0.183503419 p = 0.183503419 Simplifying p = 0.183503419Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.816496581, 0.183503419}
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