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Simplifying p(3p + 14) = 15 Reorder the terms: p(14 + 3p) = 15 (14 * p + 3p * p) = 15 (14p + 3p2) = 15 Solving 14p + 3p2 = 15 Solving for variable 'p'. Reorder the terms: -15 + 14p + 3p2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + 14p + 3p2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + 4.666666667p + p2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 4.666666667p + 5 + p2 = 0 + 5 Reorder the terms: -5 + 5 + 4.666666667p + p2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 4.666666667p + p2 = 0 + 5 4.666666667p + p2 = 0 + 5 Combine like terms: 0 + 5 = 5 4.666666667p + p2 = 5 The p term is 4.666666667p. Take half its coefficient (2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. 4.666666667p + 5.444444448 + p2 = 5 + 5.444444448 Reorder the terms: 5.444444448 + 4.666666667p + p2 = 5 + 5.444444448 Combine like terms: 5 + 5.444444448 = 10.444444448 5.444444448 + 4.666666667p + p2 = 10.444444448 Factor a perfect square on the left side: (p + 2.333333334)(p + 2.333333334) = 10.444444448 Calculate the square root of the right side: 3.231786572 Break this problem into two subproblems by setting (p + 2.333333334) equal to 3.231786572 and -3.231786572.Subproblem 1
p + 2.333333334 = 3.231786572 Simplifying p + 2.333333334 = 3.231786572 Reorder the terms: 2.333333334 + p = 3.231786572 Solving 2.333333334 + p = 3.231786572 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + p = 3.231786572 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + p = 3.231786572 + -2.333333334 p = 3.231786572 + -2.333333334 Combine like terms: 3.231786572 + -2.333333334 = 0.898453238 p = 0.898453238 Simplifying p = 0.898453238Subproblem 2
p + 2.333333334 = -3.231786572 Simplifying p + 2.333333334 = -3.231786572 Reorder the terms: 2.333333334 + p = -3.231786572 Solving 2.333333334 + p = -3.231786572 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + p = -3.231786572 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + p = -3.231786572 + -2.333333334 p = -3.231786572 + -2.333333334 Combine like terms: -3.231786572 + -2.333333334 = -5.565119906 p = -5.565119906 Simplifying p = -5.565119906Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.898453238, -5.565119906}
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