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Simplifying p2 + 3p = 1 Reorder the terms: 3p + p2 = 1 Solving 3p + p2 = 1 Solving for variable 'p'. Reorder the terms: -1 + 3p + p2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 3p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 3p + 1 + p2 = 0 + 1 Reorder the terms: -1 + 1 + 3p + p2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 3p + p2 = 0 + 1 3p + p2 = 0 + 1 Combine like terms: 0 + 1 = 1 3p + p2 = 1 The p term is 3p. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3p + 2.25 + p2 = 1 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 1 + 2.25 Combine like terms: 1 + 2.25 = 3.25 2.25 + 3p + p2 = 3.25 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (p + 1.5) equal to 1.802775638 and -1.802775638.Subproblem 1
p + 1.5 = 1.802775638 Simplifying p + 1.5 = 1.802775638 Reorder the terms: 1.5 + p = 1.802775638 Solving 1.5 + p = 1.802775638 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 1.802775638 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 1.802775638 + -1.5 p = 1.802775638 + -1.5 Combine like terms: 1.802775638 + -1.5 = 0.302775638 p = 0.302775638 Simplifying p = 0.302775638Subproblem 2
p + 1.5 = -1.802775638 Simplifying p + 1.5 = -1.802775638 Reorder the terms: 1.5 + p = -1.802775638 Solving 1.5 + p = -1.802775638 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -1.802775638 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -1.802775638 + -1.5 p = -1.802775638 + -1.5 Combine like terms: -1.802775638 + -1.5 = -3.302775638 p = -3.302775638 Simplifying p = -3.302775638Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.302775638, -3.302775638}
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