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Simplifying (5P + 1)(5P + 1) = 8 Reorder the terms: (1 + 5P)(5P + 1) = 8 Reorder the terms: (1 + 5P)(1 + 5P) = 8 Multiply (1 + 5P) * (1 + 5P) (1(1 + 5P) + 5P * (1 + 5P)) = 8 ((1 * 1 + 5P * 1) + 5P * (1 + 5P)) = 8 ((1 + 5P) + 5P * (1 + 5P)) = 8 (1 + 5P + (1 * 5P + 5P * 5P)) = 8 (1 + 5P + (5P + 25P2)) = 8 Combine like terms: 5P + 5P = 10P (1 + 10P + 25P2) = 8 Solving 1 + 10P + 25P2 = 8 Solving for variable 'P'. Reorder the terms: 1 + -8 + 10P + 25P2 = 8 + -8 Combine like terms: 1 + -8 = -7 -7 + 10P + 25P2 = 8 + -8 Combine like terms: 8 + -8 = 0 -7 + 10P + 25P2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.28 + 0.4P + P2 = 0 Move the constant term to the right: Add '0.28' to each side of the equation. -0.28 + 0.4P + 0.28 + P2 = 0 + 0.28 Reorder the terms: -0.28 + 0.28 + 0.4P + P2 = 0 + 0.28 Combine like terms: -0.28 + 0.28 = 0.00 0.00 + 0.4P + P2 = 0 + 0.28 0.4P + P2 = 0 + 0.28 Combine like terms: 0 + 0.28 = 0.28 0.4P + P2 = 0.28 The P term is 0.4P. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4P + 0.04 + P2 = 0.28 + 0.04 Reorder the terms: 0.04 + 0.4P + P2 = 0.28 + 0.04 Combine like terms: 0.28 + 0.04 = 0.32 0.04 + 0.4P + P2 = 0.32 Factor a perfect square on the left side: (P + 0.2)(P + 0.2) = 0.32 Calculate the square root of the right side: 0.565685425 Break this problem into two subproblems by setting (P + 0.2) equal to 0.565685425 and -0.565685425.Subproblem 1
P + 0.2 = 0.565685425 Simplifying P + 0.2 = 0.565685425 Reorder the terms: 0.2 + P = 0.565685425 Solving 0.2 + P = 0.565685425 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + P = 0.565685425 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + P = 0.565685425 + -0.2 P = 0.565685425 + -0.2 Combine like terms: 0.565685425 + -0.2 = 0.365685425 P = 0.365685425 Simplifying P = 0.365685425Subproblem 2
P + 0.2 = -0.565685425 Simplifying P + 0.2 = -0.565685425 Reorder the terms: 0.2 + P = -0.565685425 Solving 0.2 + P = -0.565685425 Solving for variable 'P'. Move all terms containing P to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + P = -0.565685425 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + P = -0.565685425 + -0.2 P = -0.565685425 + -0.2 Combine like terms: -0.565685425 + -0.2 = -0.765685425 P = -0.765685425 Simplifying P = -0.765685425Solution
The solution to the problem is based on the solutions from the subproblems. P = {0.365685425, -0.765685425}
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