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Simplifying 3p(5 + p) = 10 + 4p (5 * 3p + p * 3p) = 10 + 4p (15p + 3p2) = 10 + 4p Solving 15p + 3p2 = 10 + 4p Solving for variable 'p'. Reorder the terms: -10 + 15p + -4p + 3p2 = 10 + 4p + -10 + -4p Combine like terms: 15p + -4p = 11p -10 + 11p + 3p2 = 10 + 4p + -10 + -4p Reorder the terms: -10 + 11p + 3p2 = 10 + -10 + 4p + -4p Combine like terms: 10 + -10 = 0 -10 + 11p + 3p2 = 0 + 4p + -4p -10 + 11p + 3p2 = 4p + -4p Combine like terms: 4p + -4p = 0 -10 + 11p + 3p2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.333333333 + 3.666666667p + p2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 3.666666667p + 3.333333333 + p2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 3.666666667p + p2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 3.666666667p + p2 = 0 + 3.333333333 3.666666667p + p2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 3.666666667p + p2 = 3.333333333 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 = 3.333333333 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667p + p2 = 3.333333333 + 3.361111114 Combine like terms: 3.333333333 + 3.361111114 = 6.694444447 3.361111114 + 3.666666667p + p2 = 6.694444447 Factor a perfect square on the left side: (p + 1.833333334)(p + 1.833333334) = 6.694444447 Calculate the square root of the right side: 2.58736245 Break this problem into two subproblems by setting (p + 1.833333334) equal to 2.58736245 and -2.58736245.Subproblem 1
p + 1.833333334 = 2.58736245 Simplifying p + 1.833333334 = 2.58736245 Reorder the terms: 1.833333334 + p = 2.58736245 Solving 1.833333334 + p = 2.58736245 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 = 2.58736245 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + p = 2.58736245 + -1.833333334 p = 2.58736245 + -1.833333334 Combine like terms: 2.58736245 + -1.833333334 = 0.754029116 p = 0.754029116 Simplifying p = 0.754029116Subproblem 2
p + 1.833333334 = -2.58736245 Simplifying p + 1.833333334 = -2.58736245 Reorder the terms: 1.833333334 + p = -2.58736245 Solving 1.833333334 + p = -2.58736245 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 = -2.58736245 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + p = -2.58736245 + -1.833333334 p = -2.58736245 + -1.833333334 Combine like terms: -2.58736245 + -1.833333334 = -4.420695784 p = -4.420695784 Simplifying p = -4.420695784Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.754029116, -4.420695784}
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