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Simplifying 4p2 + 12p + 3 = 0 Reorder the terms: 3 + 12p + 4p2 = 0 Solving 3 + 12p + 4p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 3p + p2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 3p + -0.75 + p2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 3p + p2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 3p + p2 = 0 + -0.75 3p + p2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 3p + p2 = -0.75 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 = -0.75 + 2.25 Reorder the terms: 2.25 + 3p + p2 = -0.75 + 2.25 Combine like terms: -0.75 + 2.25 = 1.5 2.25 + 3p + p2 = 1.5 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (p + 1.5) equal to 1.224744871 and -1.224744871.Subproblem 1
p + 1.5 = 1.224744871 Simplifying p + 1.5 = 1.224744871 Reorder the terms: 1.5 + p = 1.224744871 Solving 1.5 + p = 1.224744871 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 = 1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 1.224744871 + -1.5 p = 1.224744871 + -1.5 Combine like terms: 1.224744871 + -1.5 = -0.275255129 p = -0.275255129 Simplifying p = -0.275255129Subproblem 2
p + 1.5 = -1.224744871 Simplifying p + 1.5 = -1.224744871 Reorder the terms: 1.5 + p = -1.224744871 Solving 1.5 + p = -1.224744871 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 = -1.224744871 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -1.224744871 + -1.5 p = -1.224744871 + -1.5 Combine like terms: -1.224744871 + -1.5 = -2.724744871 p = -2.724744871 Simplifying p = -2.724744871Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.275255129, -2.724744871}
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