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Simplifying 400000000p4 + -399960000p2 + 1 = 0 Reorder the terms: 1 + -399960000p2 + 400000000p4 = 0 Solving 1 + -399960000p2 + 400000000p4 = 0 Solving for variable 'p'. Begin completing the square. Divide all terms by 400000000 the coefficient of the squared term: Divide each side by '400000000'. 0.0000000025 + -0.9999p2 + p4 = 0 Move the constant term to the right: Add '-0.0000000025' to each side of the equation. 0.0000000025 + -0.9999p2 + -0.0000000025 + p4 = 0 + -0.0000000025 Reorder the terms: 0.0000000025 + -0.0000000025 + -0.9999p2 + p4 = 0 + -0.0000000025 Combine like terms: 0.0000000025 + -0.0000000025 = 0.0000000000 0.0000000000 + -0.9999p2 + p4 = 0 + -0.0000000025 -0.9999p2 + p4 = 0 + -0.0000000025 Combine like terms: 0 + -0.0000000025 = -0.0000000025 -0.9999p2 + p4 = -0.0000000025 The p term is -0.9999p2. Take half its coefficient (-0.49995). Square it (0.2499500025) and add it to both sides. Add '0.2499500025' to each side of the equation. -0.9999p2 + 0.2499500025 + p4 = -0.0000000025 + 0.2499500025 Reorder the terms: 0.2499500025 + -0.9999p2 + p4 = -0.0000000025 + 0.2499500025 Combine like terms: -0.0000000025 + 0.2499500025 = 0.24995 0.2499500025 + -0.9999p2 + p4 = 0.24995 Factor a perfect square on the left side: (p2 + -0.49995)(p2 + -0.49995) = 0.24995 Calculate the square root of the right side: 0.499949998 Break this problem into two subproblems by setting (p2 + -0.49995) equal to 0.499949998 and -0.499949998.Subproblem 1
p2 + -0.49995 = 0.499949998 Simplifying p2 + -0.49995 = 0.499949998 Reorder the terms: -0.49995 + p2 = 0.499949998 Solving -0.49995 + p2 = 0.499949998 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.49995' to each side of the equation. -0.49995 + 0.49995 + p2 = 0.499949998 + 0.49995 Combine like terms: -0.49995 + 0.49995 = 0.00000 0.00000 + p2 = 0.499949998 + 0.49995 p2 = 0.499949998 + 0.49995 Combine like terms: 0.499949998 + 0.49995 = 0.999899998 p2 = 0.999899998 Simplifying p2 = 0.999899998 Take the square root of each side: p = {-0.999949998, 0.999949998}Subproblem 2
p2 + -0.49995 = -0.499949998 Simplifying p2 + -0.49995 = -0.499949998 Reorder the terms: -0.49995 + p2 = -0.499949998 Solving -0.49995 + p2 = -0.499949998 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '0.49995' to each side of the equation. -0.49995 + 0.49995 + p2 = -0.499949998 + 0.49995 Combine like terms: -0.49995 + 0.49995 = 0.00000 0.00000 + p2 = -0.499949998 + 0.49995 p2 = -0.499949998 + 0.49995 Combine like terms: -0.499949998 + 0.49995 = 0.000000002 p2 = 0.000000002 Simplifying p2 = 0.000000002 Take the square root of each side: p = {-0.000044721, 0.000044721}Solution
The solution to the problem is based on the solutions from the subproblems. p = {-0.999949998, 0.999949998, -0.000044721, 0.000044721}
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