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Simplifying -2p(p + 3) = -3 Reorder the terms: -2p(3 + p) = -3 (3 * -2p + p * -2p) = -3 (-6p + -2p2) = -3 Solving -6p + -2p2 = -3 Solving for variable 'p'. Reorder the terms: 3 + -6p + -2p2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + -6p + -2p2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + 3p + p2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 3p + 1.5 + p2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 3p + p2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 3p + p2 = 0 + 1.5 3p + p2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 3p + p2 = 1.5 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 = 1.5 + 2.25 Reorder the terms: 2.25 + 3p + p2 = 1.5 + 2.25 Combine like terms: 1.5 + 2.25 = 3.75 2.25 + 3p + p2 = 3.75 Factor a perfect square on the left side: (p + 1.5)(p + 1.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (p + 1.5) equal to 1.936491673 and -1.936491673.Subproblem 1
p + 1.5 = 1.936491673 Simplifying p + 1.5 = 1.936491673 Reorder the terms: 1.5 + p = 1.936491673 Solving 1.5 + p = 1.936491673 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 = 1.936491673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 1.936491673 + -1.5 p = 1.936491673 + -1.5 Combine like terms: 1.936491673 + -1.5 = 0.436491673 p = 0.436491673 Simplifying p = 0.436491673Subproblem 2
p + 1.5 = -1.936491673 Simplifying p + 1.5 = -1.936491673 Reorder the terms: 1.5 + p = -1.936491673 Solving 1.5 + p = -1.936491673 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 = -1.936491673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -1.936491673 + -1.5 p = -1.936491673 + -1.5 Combine like terms: -1.936491673 + -1.5 = -3.436491673 p = -3.436491673 Simplifying p = -3.436491673Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.436491673, -3.436491673}
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