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Simplifying p2(p2 + -5) = 2p2 + -3 Reorder the terms: p2(-5 + p2) = 2p2 + -3 (-5 * p2 + p2 * p2) = 2p2 + -3 (-5p2 + p4) = 2p2 + -3 Reorder the terms: -5p2 + p4 = -3 + 2p2 Solving -5p2 + p4 = -3 + 2p2 Solving for variable 'p'. Reorder the terms: 3 + -5p2 + -2p2 + p4 = -3 + 2p2 + 3 + -2p2 Combine like terms: -5p2 + -2p2 = -7p2 3 + -7p2 + p4 = -3 + 2p2 + 3 + -2p2 Reorder the terms: 3 + -7p2 + p4 = -3 + 3 + 2p2 + -2p2 Combine like terms: -3 + 3 = 0 3 + -7p2 + p4 = 0 + 2p2 + -2p2 3 + -7p2 + p4 = 2p2 + -2p2 Combine like terms: 2p2 + -2p2 = 0 3 + -7p2 + p4 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -7p2 + -3 + p4 = 0 + -3 Reorder the terms: 3 + -3 + -7p2 + p4 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -7p2 + p4 = 0 + -3 -7p2 + p4 = 0 + -3 Combine like terms: 0 + -3 = -3 -7p2 + p4 = -3 The p term is -7p2. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7p2 + 12.25 + p4 = -3 + 12.25 Reorder the terms: 12.25 + -7p2 + p4 = -3 + 12.25 Combine like terms: -3 + 12.25 = 9.25 12.25 + -7p2 + p4 = 9.25 Factor a perfect square on the left side: (p2 + -3.5)(p2 + -3.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (p2 + -3.5) equal to 3.041381265 and -3.041381265.Subproblem 1
p2 + -3.5 = 3.041381265 Simplifying p2 + -3.5 = 3.041381265 Reorder the terms: -3.5 + p2 = 3.041381265 Solving -3.5 + p2 = 3.041381265 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + p2 = 3.041381265 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + p2 = 3.041381265 + 3.5 p2 = 3.041381265 + 3.5 Combine like terms: 3.041381265 + 3.5 = 6.541381265 p2 = 6.541381265 Simplifying p2 = 6.541381265 Take the square root of each side: p = {-2.557612415, 2.557612415}Subproblem 2
p2 + -3.5 = -3.041381265 Simplifying p2 + -3.5 = -3.041381265 Reorder the terms: -3.5 + p2 = -3.041381265 Solving -3.5 + p2 = -3.041381265 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + p2 = -3.041381265 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + p2 = -3.041381265 + 3.5 p2 = -3.041381265 + 3.5 Combine like terms: -3.041381265 + 3.5 = 0.458618735 p2 = 0.458618735 Simplifying p2 = 0.458618735 Take the square root of each side: p = {-0.677213951, 0.677213951}Solution
The solution to the problem is based on the solutions from the subproblems. p = {-2.557612415, 2.557612415, -0.677213951, 0.677213951}
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