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Simplifying 3.7p2 + -7.2p + 6.9 + -5.4 + -2.2p2 + -2.1p = 0 Reorder the terms: 6.9 + -5.4 + -7.2p + -2.1p + 3.7p2 + -2.2p2 = 0 Combine like terms: 6.9 + -5.4 = 1.5 1.5 + -7.2p + -2.1p + 3.7p2 + -2.2p2 = 0 Combine like terms: -7.2p + -2.1p = -9.3p 1.5 + -9.3p + 3.7p2 + -2.2p2 = 0 Combine like terms: 3.7p2 + -2.2p2 = 1.5p2 1.5 + -9.3p + 1.5p2 = 0 Solving 1.5 + -9.3p + 1.5p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. 1 + -6.2p + p2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -6.2p + -1 + p2 = 0 + -1 Reorder the terms: 1 + -1 + -6.2p + p2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -6.2p + p2 = 0 + -1 -6.2p + p2 = 0 + -1 Combine like terms: 0 + -1 = -1 -6.2p + p2 = -1 The p term is -6.2p. Take half its coefficient (-3.1). Square it (9.61) and add it to both sides. Add '9.61' to each side of the equation. -6.2p + 9.61 + p2 = -1 + 9.61 Reorder the terms: 9.61 + -6.2p + p2 = -1 + 9.61 Combine like terms: -1 + 9.61 = 8.61 9.61 + -6.2p + p2 = 8.61 Factor a perfect square on the left side: (p + -3.1)(p + -3.1) = 8.61 Calculate the square root of the right side: 2.93428015 Break this problem into two subproblems by setting (p + -3.1) equal to 2.93428015 and -2.93428015.Subproblem 1
p + -3.1 = 2.93428015 Simplifying p + -3.1 = 2.93428015 Reorder the terms: -3.1 + p = 2.93428015 Solving -3.1 + p = 2.93428015 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + p = 2.93428015 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + p = 2.93428015 + 3.1 p = 2.93428015 + 3.1 Combine like terms: 2.93428015 + 3.1 = 6.03428015 p = 6.03428015 Simplifying p = 6.03428015Subproblem 2
p + -3.1 = -2.93428015 Simplifying p + -3.1 = -2.93428015 Reorder the terms: -3.1 + p = -2.93428015 Solving -3.1 + p = -2.93428015 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + p = -2.93428015 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + p = -2.93428015 + 3.1 p = -2.93428015 + 3.1 Combine like terms: -2.93428015 + 3.1 = 0.16571985 p = 0.16571985 Simplifying p = 0.16571985Solution
The solution to the problem is based on the solutions from the subproblems. p = {6.03428015, 0.16571985}
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