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Simplifying 7(p + 3) + 9 = 5(p + -2) * 3p Reorder the terms: 7(3 + p) + 9 = 5(p + -2) * 3p (3 * 7 + p * 7) + 9 = 5(p + -2) * 3p (21 + 7p) + 9 = 5(p + -2) * 3p Reorder the terms: 21 + 9 + 7p = 5(p + -2) * 3p Combine like terms: 21 + 9 = 30 30 + 7p = 5(p + -2) * 3p Reorder the terms: 30 + 7p = 5(-2 + p) * 3p Reorder the terms for easier multiplication: 30 + 7p = 5 * 3p(-2 + p) Multiply 5 * 3 30 + 7p = 15p(-2 + p) 30 + 7p = (-2 * 15p + p * 15p) 30 + 7p = (-30p + 15p2) Solving 30 + 7p = -30p + 15p2 Solving for variable 'p'. Combine like terms: 7p + 30p = 37p 30 + 37p + -15p2 = -30p + 15p2 + 30p + -15p2 Reorder the terms: 30 + 37p + -15p2 = -30p + 30p + 15p2 + -15p2 Combine like terms: -30p + 30p = 0 30 + 37p + -15p2 = 0 + 15p2 + -15p2 30 + 37p + -15p2 = 15p2 + -15p2 Combine like terms: 15p2 + -15p2 = 0 30 + 37p + -15p2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -2 + -2.466666667p + p2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -2.466666667p + 2 + p2 = 0 + 2 Reorder the terms: -2 + 2 + -2.466666667p + p2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2.466666667p + p2 = 0 + 2 -2.466666667p + p2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2.466666667p + p2 = 2 The p term is -2.466666667p. Take half its coefficient (-1.233333334). Square it (1.521111113) and add it to both sides. Add '1.521111113' to each side of the equation. -2.466666667p + 1.521111113 + p2 = 2 + 1.521111113 Reorder the terms: 1.521111113 + -2.466666667p + p2 = 2 + 1.521111113 Combine like terms: 2 + 1.521111113 = 3.521111113 1.521111113 + -2.466666667p + p2 = 3.521111113 Factor a perfect square on the left side: (p + -1.233333334)(p + -1.233333334) = 3.521111113 Calculate the square root of the right side: 1.876462393 Break this problem into two subproblems by setting (p + -1.233333334) equal to 1.876462393 and -1.876462393.Subproblem 1
p + -1.233333334 = 1.876462393 Simplifying p + -1.233333334 = 1.876462393 Reorder the terms: -1.233333334 + p = 1.876462393 Solving -1.233333334 + p = 1.876462393 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + p = 1.876462393 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + p = 1.876462393 + 1.233333334 p = 1.876462393 + 1.233333334 Combine like terms: 1.876462393 + 1.233333334 = 3.109795727 p = 3.109795727 Simplifying p = 3.109795727Subproblem 2
p + -1.233333334 = -1.876462393 Simplifying p + -1.233333334 = -1.876462393 Reorder the terms: -1.233333334 + p = -1.876462393 Solving -1.233333334 + p = -1.876462393 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.233333334' to each side of the equation. -1.233333334 + 1.233333334 + p = -1.876462393 + 1.233333334 Combine like terms: -1.233333334 + 1.233333334 = 0.000000000 0.000000000 + p = -1.876462393 + 1.233333334 p = -1.876462393 + 1.233333334 Combine like terms: -1.876462393 + 1.233333334 = -0.643129059 p = -0.643129059 Simplifying p = -0.643129059Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.109795727, -0.643129059}
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