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Simplifying 9p2 = 2 + 6p Solving 9p2 = 2 + 6p Solving for variable 'p'. Reorder the terms: -2 + -6p + 9p2 = 2 + 6p + -2 + -6p Reorder the terms: -2 + -6p + 9p2 = 2 + -2 + 6p + -6p Combine like terms: 2 + -2 = 0 -2 + -6p + 9p2 = 0 + 6p + -6p -2 + -6p + 9p2 = 6p + -6p Combine like terms: 6p + -6p = 0 -2 + -6p + 9p2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.2222222222 + -0.6666666667p + p2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + -0.6666666667p + 0.2222222222 + p2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + -0.6666666667p + p2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + -0.6666666667p + p2 = 0 + 0.2222222222 -0.6666666667p + p2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 -0.6666666667p + p2 = 0.2222222222 The p term is -0.6666666667p. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667p + 0.1111111112 + p2 = 0.2222222222 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667p + p2 = 0.2222222222 + 0.1111111112 Combine like terms: 0.2222222222 + 0.1111111112 = 0.3333333334 0.1111111112 + -0.6666666667p + p2 = 0.3333333334 Factor a perfect square on the left side: (p + -0.3333333334)(p + -0.3333333334) = 0.3333333334 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (p + -0.3333333334) equal to 0.577350269 and -0.577350269.Subproblem 1
p + -0.3333333334 = 0.577350269 Simplifying p + -0.3333333334 = 0.577350269 Reorder the terms: -0.3333333334 + p = 0.577350269 Solving -0.3333333334 + p = 0.577350269 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + p = 0.577350269 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + p = 0.577350269 + 0.3333333334 p = 0.577350269 + 0.3333333334 Combine like terms: 0.577350269 + 0.3333333334 = 0.9106836024 p = 0.9106836024 Simplifying p = 0.9106836024Subproblem 2
p + -0.3333333334 = -0.577350269 Simplifying p + -0.3333333334 = -0.577350269 Reorder the terms: -0.3333333334 + p = -0.577350269 Solving -0.3333333334 + p = -0.577350269 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + p = -0.577350269 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + p = -0.577350269 + 0.3333333334 p = -0.577350269 + 0.3333333334 Combine like terms: -0.577350269 + 0.3333333334 = -0.2440169356 p = -0.2440169356 Simplifying p = -0.2440169356Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.9106836024, -0.2440169356}
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