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Simplifying 4.05 + 14.40 = 4.50p(p + 3) Combine like terms: 4.05 + 14.40 = 18.45 18.45 = 4.50p(p + 3) Reorder the terms: 18.45 = 4.50p(3 + p) 18.45 = (3 * 4.50p + p * 4.50p) 18.45 = (13.5p + 4.50p2) Solving 18.45 = 13.5p + 4.50p2 Solving for variable 'p'. Reorder the terms: 18.45 + -13.5p + -4.50p2 = 13.5p + -13.5p + 4.50p2 + -4.50p2 Combine like terms: 13.5p + -13.5p = 0.0 18.45 + -13.5p + -4.50p2 = 0.0 + 4.50p2 + -4.50p2 18.45 + -13.5p + -4.50p2 = 4.50p2 + -4.50p2 Combine like terms: 4.50p2 + -4.50p2 = 0.00 18.45 + -13.5p + -4.50p2 = 0.00 Begin completing the square. Divide all terms by -4.50 the coefficient of the squared term: Divide each side by '-4.50'. -4.1 + 3p + p2 = 0 Move the constant term to the right: Add '4.1' to each side of the equation. -4.1 + 3p + 4.1 + p2 = 0 + 4.1 Reorder the terms: -4.1 + 4.1 + 3p + p2 = 0 + 4.1 Combine like terms: -4.1 + 4.1 = 0.0 0.0 + 3p + p2 = 0 + 4.1 3p + p2 = 0 + 4.1 Combine like terms: 0 + 4.1 = 4.1 3p + p2 = 4.1 The p term is 3p. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3p + 2.25 + p2 = 4.1 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 4.1 + 2.25 Combine like terms: 4.1 + 2.25 = 6.35 2.25 + 3p + p2 = 6.35 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 6.35 Calculate the square root of the right side: 2.519920634 Break this problem into two subproblems by setting (p + 1.5) equal to 2.519920634 and -2.519920634.Subproblem 1
p + 1.5 = 2.519920634 Simplifying p + 1.5 = 2.519920634 Reorder the terms: 1.5 + p = 2.519920634 Solving 1.5 + p = 2.519920634 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 2.519920634 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 2.519920634 + -1.5 p = 2.519920634 + -1.5 Combine like terms: 2.519920634 + -1.5 = 1.019920634 p = 1.019920634 Simplifying p = 1.019920634Subproblem 2
p + 1.5 = -2.519920634 Simplifying p + 1.5 = -2.519920634 Reorder the terms: 1.5 + p = -2.519920634 Solving 1.5 + p = -2.519920634 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -2.519920634 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -2.519920634 + -1.5 p = -2.519920634 + -1.5 Combine like terms: -2.519920634 + -1.5 = -4.019920634 p = -4.019920634 Simplifying p = -4.019920634Solution
The solution to the problem is based on the solutions from the subproblems. p = {1.019920634, -4.019920634}
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