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Simplifying 3p(2p + 10) + (5 + -1p)(p + -4) = 0 Reorder the terms: 3p(10 + 2p) + (5 + -1p)(p + -4) = 0 (10 * 3p + 2p * 3p) + (5 + -1p)(p + -4) = 0 (30p + 6p2) + (5 + -1p)(p + -4) = 0 Reorder the terms: 30p + 6p2 + (5 + -1p)(-4 + p) = 0 Multiply (5 + -1p) * (-4 + p) 30p + 6p2 + (5(-4 + p) + -1p * (-4 + p)) = 0 30p + 6p2 + ((-4 * 5 + p * 5) + -1p * (-4 + p)) = 0 30p + 6p2 + ((-20 + 5p) + -1p * (-4 + p)) = 0 30p + 6p2 + (-20 + 5p + (-4 * -1p + p * -1p)) = 0 30p + 6p2 + (-20 + 5p + (4p + -1p2)) = 0 Combine like terms: 5p + 4p = 9p 30p + 6p2 + (-20 + 9p + -1p2) = 0 Reorder the terms: -20 + 30p + 9p + 6p2 + -1p2 = 0 Combine like terms: 30p + 9p = 39p -20 + 39p + 6p2 + -1p2 = 0 Combine like terms: 6p2 + -1p2 = 5p2 -20 + 39p + 5p2 = 0 Solving -20 + 39p + 5p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4 + 7.8p + p2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 7.8p + 4 + p2 = 0 + 4 Reorder the terms: -4 + 4 + 7.8p + p2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 7.8p + p2 = 0 + 4 7.8p + p2 = 0 + 4 Combine like terms: 0 + 4 = 4 7.8p + p2 = 4 The p term is 7.8p. Take half its coefficient (3.9). Square it (15.21) and add it to both sides. Add '15.21' to each side of the equation. 7.8p + 15.21 + p2 = 4 + 15.21 Reorder the terms: 15.21 + 7.8p + p2 = 4 + 15.21 Combine like terms: 4 + 15.21 = 19.21 15.21 + 7.8p + p2 = 19.21 Factor a perfect square on the left side: (p + 3.9)(p + 3.9) = 19.21 Calculate the square root of the right side: 4.3829214 Break this problem into two subproblems by setting (p + 3.9) equal to 4.3829214 and -4.3829214.Subproblem 1
p + 3.9 = 4.3829214 Simplifying p + 3.9 = 4.3829214 Reorder the terms: 3.9 + p = 4.3829214 Solving 3.9 + p = 4.3829214 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3.9' to each side of the equation. 3.9 + -3.9 + p = 4.3829214 + -3.9 Combine like terms: 3.9 + -3.9 = 0.0 0.0 + p = 4.3829214 + -3.9 p = 4.3829214 + -3.9 Combine like terms: 4.3829214 + -3.9 = 0.4829214 p = 0.4829214 Simplifying p = 0.4829214Subproblem 2
p + 3.9 = -4.3829214 Simplifying p + 3.9 = -4.3829214 Reorder the terms: 3.9 + p = -4.3829214 Solving 3.9 + p = -4.3829214 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3.9' to each side of the equation. 3.9 + -3.9 + p = -4.3829214 + -3.9 Combine like terms: 3.9 + -3.9 = 0.0 0.0 + p = -4.3829214 + -3.9 p = -4.3829214 + -3.9 Combine like terms: -4.3829214 + -3.9 = -8.2829214 p = -8.2829214 Simplifying p = -8.2829214Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.4829214, -8.2829214}
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