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Simplifying 9p2 + -12p + 2 = 0 Reorder the terms: 2 + -12p + 9p2 = 0 Solving 2 + -12p + 9p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.2222222222 + -1.333333333p + p2 = 0 Move the constant term to the right: Add '-0.2222222222' to each side of the equation. 0.2222222222 + -1.333333333p + -0.2222222222 + p2 = 0 + -0.2222222222 Reorder the terms: 0.2222222222 + -0.2222222222 + -1.333333333p + p2 = 0 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + -1.333333333p + p2 = 0 + -0.2222222222 -1.333333333p + p2 = 0 + -0.2222222222 Combine like terms: 0 + -0.2222222222 = -0.2222222222 -1.333333333p + p2 = -0.2222222222 The p term is -1.333333333p. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333p + 0.4444444442 + p2 = -0.2222222222 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333p + p2 = -0.2222222222 + 0.4444444442 Combine like terms: -0.2222222222 + 0.4444444442 = 0.222222222 0.4444444442 + -1.333333333p + p2 = 0.222222222 Factor a perfect square on the left side: (p + -0.6666666665)(p + -0.6666666665) = 0.222222222 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (p + -0.6666666665) equal to 0.471404521 and -0.471404521.Subproblem 1
p + -0.6666666665 = 0.471404521 Simplifying p + -0.6666666665 = 0.471404521 Reorder the terms: -0.6666666665 + p = 0.471404521 Solving -0.6666666665 + p = 0.471404521 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + p = 0.471404521 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + p = 0.471404521 + 0.6666666665 p = 0.471404521 + 0.6666666665 Combine like terms: 0.471404521 + 0.6666666665 = 1.1380711875 p = 1.1380711875 Simplifying p = 1.1380711875Subproblem 2
p + -0.6666666665 = -0.471404521 Simplifying p + -0.6666666665 = -0.471404521 Reorder the terms: -0.6666666665 + p = -0.471404521 Solving -0.6666666665 + p = -0.471404521 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + p = -0.471404521 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + p = -0.471404521 + 0.6666666665 p = -0.471404521 + 0.6666666665 Combine like terms: -0.471404521 + 0.6666666665 = 0.1952621455 p = 0.1952621455 Simplifying p = 0.1952621455Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.1380711875, 0.1952621455}
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