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Simplifying (p + 2) + 8p(3p + 6) = 80 Reorder the terms: (2 + p) + 8p(3p + 6) = 80 Remove parenthesis around (2 + p) 2 + p + 8p(3p + 6) = 80 Reorder the terms: 2 + p + 8p(6 + 3p) = 80 2 + p + (6 * 8p + 3p * 8p) = 80 2 + p + (48p + 24p2) = 80 Combine like terms: p + 48p = 49p 2 + 49p + 24p2 = 80 Solving 2 + 49p + 24p2 = 80 Solving for variable 'p'. Reorder the terms: 2 + -80 + 49p + 24p2 = 80 + -80 Combine like terms: 2 + -80 = -78 -78 + 49p + 24p2 = 80 + -80 Combine like terms: 80 + -80 = 0 -78 + 49p + 24p2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -3.25 + 2.041666667p + p2 = 0 Move the constant term to the right: Add '3.25' to each side of the equation. -3.25 + 2.041666667p + 3.25 + p2 = 0 + 3.25 Reorder the terms: -3.25 + 3.25 + 2.041666667p + p2 = 0 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + 2.041666667p + p2 = 0 + 3.25 2.041666667p + p2 = 0 + 3.25 Combine like terms: 0 + 3.25 = 3.25 2.041666667p + p2 = 3.25 The p term is 2.041666667p. Take half its coefficient (1.020833334). Square it (1.042100696) and add it to both sides. Add '1.042100696' to each side of the equation. 2.041666667p + 1.042100696 + p2 = 3.25 + 1.042100696 Reorder the terms: 1.042100696 + 2.041666667p + p2 = 3.25 + 1.042100696 Combine like terms: 3.25 + 1.042100696 = 4.292100696 1.042100696 + 2.041666667p + p2 = 4.292100696 Factor a perfect square on the left side: (p + 1.020833334)(p + 1.020833334) = 4.292100696 Calculate the square root of the right side: 2.071738568 Break this problem into two subproblems by setting (p + 1.020833334) equal to 2.071738568 and -2.071738568.Subproblem 1
p + 1.020833334 = 2.071738568 Simplifying p + 1.020833334 = 2.071738568 Reorder the terms: 1.020833334 + p = 2.071738568 Solving 1.020833334 + p = 2.071738568 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + p = 2.071738568 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + p = 2.071738568 + -1.020833334 p = 2.071738568 + -1.020833334 Combine like terms: 2.071738568 + -1.020833334 = 1.050905234 p = 1.050905234 Simplifying p = 1.050905234Subproblem 2
p + 1.020833334 = -2.071738568 Simplifying p + 1.020833334 = -2.071738568 Reorder the terms: 1.020833334 + p = -2.071738568 Solving 1.020833334 + p = -2.071738568 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + p = -2.071738568 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + p = -2.071738568 + -1.020833334 p = -2.071738568 + -1.020833334 Combine like terms: -2.071738568 + -1.020833334 = -3.092571902 p = -3.092571902 Simplifying p = -3.092571902Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.050905234, -3.092571902}
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