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Simplifying 2p + 3p(p + 3) = 21 Reorder the terms: 2p + 3p(3 + p) = 21 2p + (3 * 3p + p * 3p) = 21 2p + (9p + 3p2) = 21 Combine like terms: 2p + 9p = 11p 11p + 3p2 = 21 Solving 11p + 3p2 = 21 Solving for variable 'p'. Reorder the terms: -21 + 11p + 3p2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + 11p + 3p2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7 + 3.666666667p + p2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 3.666666667p + 7 + p2 = 0 + 7 Reorder the terms: -7 + 7 + 3.666666667p + p2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 3.666666667p + p2 = 0 + 7 3.666666667p + p2 = 0 + 7 Combine like terms: 0 + 7 = 7 3.666666667p + p2 = 7 The p term is 3.666666667p. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667p + 3.361111114 + p2 = 7 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667p + p2 = 7 + 3.361111114 Combine like terms: 7 + 3.361111114 = 10.361111114 3.361111114 + 3.666666667p + p2 = 10.361111114 Factor a perfect square on the left side: (p + 1.833333334)(p + 1.833333334) = 10.361111114 Calculate the square root of the right side: 3.218867986 Break this problem into two subproblems by setting (p + 1.833333334) equal to 3.218867986 and -3.218867986.Subproblem 1
p + 1.833333334 = 3.218867986 Simplifying p + 1.833333334 = 3.218867986 Reorder the terms: 1.833333334 + p = 3.218867986 Solving 1.833333334 + p = 3.218867986 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + p = 3.218867986 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + p = 3.218867986 + -1.833333334 p = 3.218867986 + -1.833333334 Combine like terms: 3.218867986 + -1.833333334 = 1.385534652 p = 1.385534652 Simplifying p = 1.385534652Subproblem 2
p + 1.833333334 = -3.218867986 Simplifying p + 1.833333334 = -3.218867986 Reorder the terms: 1.833333334 + p = -3.218867986 Solving 1.833333334 + p = -3.218867986 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + p = -3.218867986 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + p = -3.218867986 + -1.833333334 p = -3.218867986 + -1.833333334 Combine like terms: -3.218867986 + -1.833333334 = -5.05220132 p = -5.05220132 Simplifying p = -5.05220132Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.385534652, -5.05220132}
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