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Simplifying 3p * p + -12p + 1 = 0 Multiply p * p 3p2 + -12p + 1 = 0 Reorder the terms: 1 + -12p + 3p2 = 0 Solving 1 + -12p + 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 + -4p + p2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -4p + -0.3333333333 + p2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -4p + p2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -4p + p2 = 0 + -0.3333333333 -4p + p2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -4p + p2 = -0.3333333333 The p term is -4p. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4p + 4 + p2 = -0.3333333333 + 4 Reorder the terms: 4 + -4p + p2 = -0.3333333333 + 4 Combine like terms: -0.3333333333 + 4 = 3.6666666667 4 + -4p + p2 = 3.6666666667 Factor a perfect square on the left side: (p + -2)(p + -2) = 3.6666666667 Calculate the square root of the right side: 1.914854216 Break this problem into two subproblems by setting (p + -2) equal to 1.914854216 and -1.914854216.Subproblem 1
p + -2 = 1.914854216 Simplifying p + -2 = 1.914854216 Reorder the terms: -2 + p = 1.914854216 Solving -2 + p = 1.914854216 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + p = 1.914854216 + 2 Combine like terms: -2 + 2 = 0 0 + p = 1.914854216 + 2 p = 1.914854216 + 2 Combine like terms: 1.914854216 + 2 = 3.914854216 p = 3.914854216 Simplifying p = 3.914854216Subproblem 2
p + -2 = -1.914854216 Simplifying p + -2 = -1.914854216 Reorder the terms: -2 + p = -1.914854216 Solving -2 + p = -1.914854216 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + p = -1.914854216 + 2 Combine like terms: -2 + 2 = 0 0 + p = -1.914854216 + 2 p = -1.914854216 + 2 Combine like terms: -1.914854216 + 2 = 0.085145784 p = 0.085145784 Simplifying p = 0.085145784Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.914854216, 0.085145784}
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