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Simplifying 6p2 + p + -32 = 0 Reorder the terms: -32 + p + 6p2 = 0 Solving -32 + p + 6p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.333333333 + 0.1666666667p + p2 = 0 Move the constant term to the right: Add '5.333333333' to each side of the equation. -5.333333333 + 0.1666666667p + 5.333333333 + p2 = 0 + 5.333333333 Reorder the terms: -5.333333333 + 5.333333333 + 0.1666666667p + p2 = 0 + 5.333333333 Combine like terms: -5.333333333 + 5.333333333 = 0.000000000 0.000000000 + 0.1666666667p + p2 = 0 + 5.333333333 0.1666666667p + p2 = 0 + 5.333333333 Combine like terms: 0 + 5.333333333 = 5.333333333 0.1666666667p + p2 = 5.333333333 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. 0.1666666667p + 0.25 + p2 = 5.333333333 + 0.25 Reorder the terms: 0.25 + 0.1666666667p + p2 = 5.333333333 + 0.25 Combine like terms: 5.333333333 + 0.25 = 5.583333333 0.25 + 0.1666666667p + p2 = 5.583333333 Factor a perfect square on the left side: (p + 0.5)(p + 0.5) = 5.583333333 Calculate the square root of the right side: 2.362907813 Break this problem into two subproblems by setting (p + 0.5) equal to 2.362907813 and -2.362907813.Subproblem 1
p + 0.5 = 2.362907813 Simplifying p + 0.5 = 2.362907813 Reorder the terms: 0.5 + p = 2.362907813 Solving 0.5 + p = 2.362907813 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 = 2.362907813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = 2.362907813 + -0.5 p = 2.362907813 + -0.5 Combine like terms: 2.362907813 + -0.5 = 1.862907813 p = 1.862907813 Simplifying p = 1.862907813Subproblem 2
p + 0.5 = -2.362907813 Simplifying p + 0.5 = -2.362907813 Reorder the terms: 0.5 + p = -2.362907813 Solving 0.5 + p = -2.362907813 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 = -2.362907813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + p = -2.362907813 + -0.5 p = -2.362907813 + -0.5 Combine like terms: -2.362907813 + -0.5 = -2.862907813 p = -2.862907813 Simplifying p = -2.862907813Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.862907813, -2.862907813}
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