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Simplifying (p + 2)(p + -5) = 2(p + 3) Reorder the terms: (2 + p)(p + -5) = 2(p + 3) Reorder the terms: (2 + p)(-5 + p) = 2(p + 3) Multiply (2 + p) * (-5 + p) (2(-5 + p) + p(-5 + p)) = 2(p + 3) ((-5 * 2 + p * 2) + p(-5 + p)) = 2(p + 3) ((-10 + 2p) + p(-5 + p)) = 2(p + 3) (-10 + 2p + (-5 * p + p * p)) = 2(p + 3) (-10 + 2p + (-5p + p2)) = 2(p + 3) Combine like terms: 2p + -5p = -3p (-10 + -3p + p2) = 2(p + 3) Reorder the terms: -10 + -3p + p2 = 2(3 + p) -10 + -3p + p2 = (3 * 2 + p * 2) -10 + -3p + p2 = (6 + 2p) Solving -10 + -3p + p2 = 6 + 2p Solving for variable 'p'. Reorder the terms: -10 + -6 + -3p + -2p + p2 = 6 + 2p + -6 + -2p Combine like terms: -10 + -6 = -16 -16 + -3p + -2p + p2 = 6 + 2p + -6 + -2p Combine like terms: -3p + -2p = -5p -16 + -5p + p2 = 6 + 2p + -6 + -2p Reorder the terms: -16 + -5p + p2 = 6 + -6 + 2p + -2p Combine like terms: 6 + -6 = 0 -16 + -5p + p2 = 0 + 2p + -2p -16 + -5p + p2 = 2p + -2p Combine like terms: 2p + -2p = 0 -16 + -5p + p2 = 0 Begin completing the square. Move the constant term to the right: Add '16' to each side of the equation. -16 + -5p + 16 + p2 = 0 + 16 Reorder the terms: -16 + 16 + -5p + p2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + -5p + p2 = 0 + 16 -5p + p2 = 0 + 16 Combine like terms: 0 + 16 = 16 -5p + p2 = 16 The p term is -5p. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5p + 6.25 + p2 = 16 + 6.25 Reorder the terms: 6.25 + -5p + p2 = 16 + 6.25 Combine like terms: 16 + 6.25 = 22.25 6.25 + -5p + p2 = 22.25 Factor a perfect square on the left side: (p + -2.5)(p + -2.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (p + -2.5) equal to 4.716990566 and -4.716990566.Subproblem 1
p + -2.5 = 4.716990566 Simplifying p + -2.5 = 4.716990566 Reorder the terms: -2.5 + p = 4.716990566 Solving -2.5 + p = 4.716990566 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = 4.716990566 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = 4.716990566 + 2.5 p = 4.716990566 + 2.5 Combine like terms: 4.716990566 + 2.5 = 7.216990566 p = 7.216990566 Simplifying p = 7.216990566Subproblem 2
p + -2.5 = -4.716990566 Simplifying p + -2.5 = -4.716990566 Reorder the terms: -2.5 + p = -4.716990566 Solving -2.5 + p = -4.716990566 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = -4.716990566 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = -4.716990566 + 2.5 p = -4.716990566 + 2.5 Combine like terms: -4.716990566 + 2.5 = -2.216990566 p = -2.216990566 Simplifying p = -2.216990566Solution
The solution to the problem is based on the solutions from the subproblems. p = {7.216990566, -2.216990566}
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