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Simplifying (7p + 8)(2p + -3) = 9p + 5 Reorder the terms: (8 + 7p)(2p + -3) = 9p + 5 Reorder the terms: (8 + 7p)(-3 + 2p) = 9p + 5 Multiply (8 + 7p) * (-3 + 2p) (8(-3 + 2p) + 7p * (-3 + 2p)) = 9p + 5 ((-3 * 8 + 2p * 8) + 7p * (-3 + 2p)) = 9p + 5 ((-24 + 16p) + 7p * (-3 + 2p)) = 9p + 5 (-24 + 16p + (-3 * 7p + 2p * 7p)) = 9p + 5 (-24 + 16p + (-21p + 14p2)) = 9p + 5 Combine like terms: 16p + -21p = -5p (-24 + -5p + 14p2) = 9p + 5 Reorder the terms: -24 + -5p + 14p2 = 5 + 9p Solving -24 + -5p + 14p2 = 5 + 9p Solving for variable 'p'. Reorder the terms: -24 + -5 + -5p + -9p + 14p2 = 5 + 9p + -5 + -9p Combine like terms: -24 + -5 = -29 -29 + -5p + -9p + 14p2 = 5 + 9p + -5 + -9p Combine like terms: -5p + -9p = -14p -29 + -14p + 14p2 = 5 + 9p + -5 + -9p Reorder the terms: -29 + -14p + 14p2 = 5 + -5 + 9p + -9p Combine like terms: 5 + -5 = 0 -29 + -14p + 14p2 = 0 + 9p + -9p -29 + -14p + 14p2 = 9p + -9p Combine like terms: 9p + -9p = 0 -29 + -14p + 14p2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -2.071428571 + -1p + p2 = 0 Move the constant term to the right: Add '2.071428571' to each side of the equation. -2.071428571 + -1p + 2.071428571 + p2 = 0 + 2.071428571 Reorder the terms: -2.071428571 + 2.071428571 + -1p + p2 = 0 + 2.071428571 Combine like terms: -2.071428571 + 2.071428571 = 0.000000000 0.000000000 + -1p + p2 = 0 + 2.071428571 -1p + p2 = 0 + 2.071428571 Combine like terms: 0 + 2.071428571 = 2.071428571 -1p + p2 = 2.071428571 The p term is -1p. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1p + 0.25 + p2 = 2.071428571 + 0.25 Reorder the terms: 0.25 + -1p + p2 = 2.071428571 + 0.25 Combine like terms: 2.071428571 + 0.25 = 2.321428571 0.25 + -1p + p2 = 2.321428571 Factor a perfect square on the left side: (p + -0.5)(p + -0.5) = 2.321428571 Calculate the square root of the right side: 1.5236235 Break this problem into two subproblems by setting (p + -0.5) equal to 1.5236235 and -1.5236235.Subproblem 1
p + -0.5 = 1.5236235 Simplifying p + -0.5 = 1.5236235 Reorder the terms: -0.5 + p = 1.5236235 Solving -0.5 + p = 1.5236235 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = 1.5236235 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = 1.5236235 + 0.5 p = 1.5236235 + 0.5 Combine like terms: 1.5236235 + 0.5 = 2.0236235 p = 2.0236235 Simplifying p = 2.0236235Subproblem 2
p + -0.5 = -1.5236235 Simplifying p + -0.5 = -1.5236235 Reorder the terms: -0.5 + p = -1.5236235 Solving -0.5 + p = -1.5236235 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + p = -1.5236235 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + p = -1.5236235 + 0.5 p = -1.5236235 + 0.5 Combine like terms: -1.5236235 + 0.5 = -1.0236235 p = -1.0236235 Simplifying p = -1.0236235Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.0236235, -1.0236235}
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