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Simplifying 9p2 + -24p + 11 = 0 Reorder the terms: 11 + -24p + 9p2 = 0 Solving 11 + -24p + 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'. 1.222222222 + -2.666666667p + p2 = 0 Move the constant term to the right: Add '-1.222222222' to each side of the equation. 1.222222222 + -2.666666667p + -1.222222222 + p2 = 0 + -1.222222222 Reorder the terms: 1.222222222 + -1.222222222 + -2.666666667p + p2 = 0 + -1.222222222 Combine like terms: 1.222222222 + -1.222222222 = 0.000000000 0.000000000 + -2.666666667p + p2 = 0 + -1.222222222 -2.666666667p + p2 = 0 + -1.222222222 Combine like terms: 0 + -1.222222222 = -1.222222222 -2.666666667p + p2 = -1.222222222 The p term is -2.666666667p. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667p + 1.777777780 + p2 = -1.222222222 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667p + p2 = -1.222222222 + 1.777777780 Combine like terms: -1.222222222 + 1.777777780 = 0.555555558 1.777777780 + -2.666666667p + p2 = 0.555555558 Factor a perfect square on the left side: (p + -1.333333334)(p + -1.333333334) = 0.555555558 Calculate the square root of the right side: 0.745355994 Break this problem into two subproblems by setting (p + -1.333333334) equal to 0.745355994 and -0.745355994.Subproblem 1
p + -1.333333334 = 0.745355994 Simplifying p + -1.333333334 = 0.745355994 Reorder the terms: -1.333333334 + p = 0.745355994 Solving -1.333333334 + p = 0.745355994 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + p = 0.745355994 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + p = 0.745355994 + 1.333333334 p = 0.745355994 + 1.333333334 Combine like terms: 0.745355994 + 1.333333334 = 2.078689328 p = 2.078689328 Simplifying p = 2.078689328Subproblem 2
p + -1.333333334 = -0.745355994 Simplifying p + -1.333333334 = -0.745355994 Reorder the terms: -1.333333334 + p = -0.745355994 Solving -1.333333334 + p = -0.745355994 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + p = -0.745355994 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + p = -0.745355994 + 1.333333334 p = -0.745355994 + 1.333333334 Combine like terms: -0.745355994 + 1.333333334 = 0.58797734 p = 0.58797734 Simplifying p = 0.58797734Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.078689328, 0.58797734}
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