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Simplifying 7 = 2p(p) + -5p + 8 Multiply p * p 7 = 2p2 + -5p + 8 Reorder the terms: 7 = 8 + -5p + 2p2 Solving 7 = 8 + -5p + 2p2 Solving for variable 'p'. Combine like terms: 7 + -8 = -1 -1 + 5p + -2p2 = 8 + -5p + 2p2 + -8 + 5p + -2p2 Reorder the terms: -1 + 5p + -2p2 = 8 + -8 + -5p + 5p + 2p2 + -2p2 Combine like terms: 8 + -8 = 0 -1 + 5p + -2p2 = 0 + -5p + 5p + 2p2 + -2p2 -1 + 5p + -2p2 = -5p + 5p + 2p2 + -2p2 Combine like terms: -5p + 5p = 0 -1 + 5p + -2p2 = 0 + 2p2 + -2p2 -1 + 5p + -2p2 = 2p2 + -2p2 Combine like terms: 2p2 + -2p2 = 0 -1 + 5p + -2p2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -2.5p + p2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2.5p + -0.5 + p2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2.5p + p2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2.5p + p2 = 0 + -0.5 -2.5p + p2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2.5p + p2 = -0.5 The p term is -2.5p. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5p + 1.5625 + p2 = -0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5p + p2 = -0.5 + 1.5625 Combine like terms: -0.5 + 1.5625 = 1.0625 1.5625 + -2.5p + p2 = 1.0625 Factor a perfect square on the left side: (p + -1.25)(p + -1.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (p + -1.25) equal to 1.030776406 and -1.030776406.Subproblem 1
p + -1.25 = 1.030776406 Simplifying p + -1.25 = 1.030776406 Reorder the terms: -1.25 + p = 1.030776406 Solving -1.25 + p = 1.030776406 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + p = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + p = 1.030776406 + 1.25 p = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 p = 2.280776406 Simplifying p = 2.280776406Subproblem 2
p + -1.25 = -1.030776406 Simplifying p + -1.25 = -1.030776406 Reorder the terms: -1.25 + p = -1.030776406 Solving -1.25 + p = -1.030776406 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + p = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + p = -1.030776406 + 1.25 p = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 p = 0.219223594 Simplifying p = 0.219223594Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.280776406, 0.219223594}
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