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Simplifying 2.5p(2p + 5) = 5p + 12.5 Reorder the terms: 2.5p(5 + 2p) = 5p + 12.5 (5 * 2.5p + 2p * 2.5p) = 5p + 12.5 (12.5p + 5p2) = 5p + 12.5 Reorder the terms: 12.5p + 5p2 = 12.5 + 5p Solving 12.5p + 5p2 = 12.5 + 5p Solving for variable 'p'. Reorder the terms: -12.5 + 12.5p + -5p + 5p2 = 12.5 + 5p + -12.5 + -5p Combine like terms: 12.5p + -5p = 7.5p -12.5 + 7.5p + 5p2 = 12.5 + 5p + -12.5 + -5p Reorder the terms: -12.5 + 7.5p + 5p2 = 12.5 + -12.5 + 5p + -5p Combine like terms: 12.5 + -12.5 = 0.0 -12.5 + 7.5p + 5p2 = 0.0 + 5p + -5p -12.5 + 7.5p + 5p2 = 5p + -5p Combine like terms: 5p + -5p = 0 -12.5 + 7.5p + 5p2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.5 + 1.5p + p2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 1.5p + 2.5 + p2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 1.5p + p2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 1.5p + p2 = 0 + 2.5 1.5p + p2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 1.5p + p2 = 2.5 The p term is 1.5p. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5p + 0.5625 + p2 = 2.5 + 0.5625 Reorder the terms: 0.5625 + 1.5p + p2 = 2.5 + 0.5625 Combine like terms: 2.5 + 0.5625 = 3.0625 0.5625 + 1.5p + p2 = 3.0625 Factor a perfect square on the left side: (p + 0.75)(p + 0.75) = 3.0625 Calculate the square root of the right side: 1.75 Break this problem into two subproblems by setting (p + 0.75) equal to 1.75 and -1.75.Subproblem 1
p + 0.75 = 1.75 Simplifying p + 0.75 = 1.75 Reorder the terms: 0.75 + p = 1.75 Solving 0.75 + p = 1.75 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = 1.75 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = 1.75 + -0.75 p = 1.75 + -0.75 Combine like terms: 1.75 + -0.75 = 1 p = 1 Simplifying p = 1Subproblem 2
p + 0.75 = -1.75 Simplifying p + 0.75 = -1.75 Reorder the terms: 0.75 + p = -1.75 Solving 0.75 + p = -1.75 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = -1.75 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = -1.75 + -0.75 p = -1.75 + -0.75 Combine like terms: -1.75 + -0.75 = -2.5 p = -2.5 Simplifying p = -2.5Solution
The solution to the problem is based on the solutions from the subproblems. p = {1, -2.5}
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