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Simplifying 4p2 + -1(2p2 + -3p + 1) = 0 Reorder the terms: 4p2 + -1(1 + -3p + 2p2) = 0 4p2 + (1 * -1 + -3p * -1 + 2p2 * -1) = 0 4p2 + (-1 + 3p + -2p2) = 0 Reorder the terms: -1 + 3p + 4p2 + -2p2 = 0 Combine like terms: 4p2 + -2p2 = 2p2 -1 + 3p + 2p2 = 0 Solving -1 + 3p + 2p2 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 1.5p + p2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.5p + 0.5 + p2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.5p + p2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.5p + p2 = 0 + 0.5 1.5p + p2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.5p + p2 = 0.5 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 = 0.5 + 0.5625 Reorder the terms: 0.5625 + 1.5p + p2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + 1.5p + p2 = 1.0625 Factor a perfect square on the left side: (p + 0.75)(p + 0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (p + 0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
p + 0.75 = 1.030776406 Simplifying p + 0.75 = 1.030776406 Reorder the terms: 0.75 + p = 1.030776406 Solving 0.75 + p = 1.030776406 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 = 1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = 1.030776406 + -0.75 p = 1.030776406 + -0.75 Combine like terms: 1.030776406 + -0.75 = 0.280776406 p = 0.280776406 Simplifying p = 0.280776406Subproblem 2
p + 0.75 = -1.030776406 Simplifying p + 0.75 = -1.030776406 Reorder the terms: 0.75 + p = -1.030776406 Solving 0.75 + p = -1.030776406 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 = -1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = -1.030776406 + -0.75 p = -1.030776406 + -0.75 Combine like terms: -1.030776406 + -0.75 = -1.780776406 p = -1.780776406 Simplifying p = -1.780776406Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.280776406, -1.780776406}
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