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Simplifying (2p + -5)(p + 0.75) = 0 Reorder the terms: (-5 + 2p)(p + 0.75) = 0 Reorder the terms: (-5 + 2p)(0.75 + p) = 0 Multiply (-5 + 2p) * (0.75 + p) (-5(0.75 + p) + 2p * (0.75 + p)) = 0 ((0.75 * -5 + p * -5) + 2p * (0.75 + p)) = 0 ((-3.75 + -5p) + 2p * (0.75 + p)) = 0 (-3.75 + -5p + (0.75 * 2p + p * 2p)) = 0 (-3.75 + -5p + (1.5p + 2p2)) = 0 Combine like terms: -5p + 1.5p = -3.5p (-3.75 + -3.5p + 2p2) = 0 Solving -3.75 + -3.5p + 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'. -1.875 + -1.75p + p2 = 0 Move the constant term to the right: Add '1.875' to each side of the equation. -1.875 + -1.75p + 1.875 + p2 = 0 + 1.875 Reorder the terms: -1.875 + 1.875 + -1.75p + p2 = 0 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + -1.75p + p2 = 0 + 1.875 -1.75p + p2 = 0 + 1.875 Combine like terms: 0 + 1.875 = 1.875 -1.75p + p2 = 1.875 The p term is -1.75p. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75p + 0.765625 + p2 = 1.875 + 0.765625 Reorder the terms: 0.765625 + -1.75p + p2 = 1.875 + 0.765625 Combine like terms: 1.875 + 0.765625 = 2.640625 0.765625 + -1.75p + p2 = 2.640625 Factor a perfect square on the left side: (p + -0.875)(p + -0.875) = 2.640625 Calculate the square root of the right side: 1.625 Break this problem into two subproblems by setting (p + -0.875) equal to 1.625 and -1.625.Subproblem 1
p + -0.875 = 1.625 Simplifying p + -0.875 = 1.625 Reorder the terms: -0.875 + p = 1.625 Solving -0.875 + p = 1.625 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + p = 1.625 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + p = 1.625 + 0.875 p = 1.625 + 0.875 Combine like terms: 1.625 + 0.875 = 2.5 p = 2.5 Simplifying p = 2.5Subproblem 2
p + -0.875 = -1.625 Simplifying p + -0.875 = -1.625 Reorder the terms: -0.875 + p = -1.625 Solving -0.875 + p = -1.625 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + p = -1.625 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + p = -1.625 + 0.875 p = -1.625 + 0.875 Combine like terms: -1.625 + 0.875 = -0.75 p = -0.75 Simplifying p = -0.75Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.5, -0.75}
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