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Simplifying 9p(9p + 10) = 25 Reorder the terms: 9p(10 + 9p) = 25 (10 * 9p + 9p * 9p) = 25 (90p + 81p2) = 25 Solving 90p + 81p2 = 25 Solving for variable 'p'. Reorder the terms: -25 + 90p + 81p2 = 25 + -25 Combine like terms: 25 + -25 = 0 -25 + 90p + 81p2 = 0 Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -0.3086419753 + 1.111111111p + p2 = 0 Move the constant term to the right: Add '0.3086419753' to each side of the equation. -0.3086419753 + 1.111111111p + 0.3086419753 + p2 = 0 + 0.3086419753 Reorder the terms: -0.3086419753 + 0.3086419753 + 1.111111111p + p2 = 0 + 0.3086419753 Combine like terms: -0.3086419753 + 0.3086419753 = 0.0000000000 0.0000000000 + 1.111111111p + p2 = 0 + 0.3086419753 1.111111111p + p2 = 0 + 0.3086419753 Combine like terms: 0 + 0.3086419753 = 0.3086419753 1.111111111p + p2 = 0.3086419753 The p term is 1.111111111p. Take half its coefficient (0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. 1.111111111p + 0.3086419752 + p2 = 0.3086419753 + 0.3086419752 Reorder the terms: 0.3086419752 + 1.111111111p + p2 = 0.3086419753 + 0.3086419752 Combine like terms: 0.3086419753 + 0.3086419752 = 0.6172839505 0.3086419752 + 1.111111111p + p2 = 0.6172839505 Factor a perfect square on the left side: (p + 0.5555555555)(p + 0.5555555555) = 0.6172839505 Calculate the square root of the right side: 0.785674201 Break this problem into two subproblems by setting (p + 0.5555555555) equal to 0.785674201 and -0.785674201.Subproblem 1
p + 0.5555555555 = 0.785674201 Simplifying p + 0.5555555555 = 0.785674201 Reorder the terms: 0.5555555555 + p = 0.785674201 Solving 0.5555555555 + p = 0.785674201 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + p = 0.785674201 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + p = 0.785674201 + -0.5555555555 p = 0.785674201 + -0.5555555555 Combine like terms: 0.785674201 + -0.5555555555 = 0.2301186455 p = 0.2301186455 Simplifying p = 0.2301186455Subproblem 2
p + 0.5555555555 = -0.785674201 Simplifying p + 0.5555555555 = -0.785674201 Reorder the terms: 0.5555555555 + p = -0.785674201 Solving 0.5555555555 + p = -0.785674201 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + p = -0.785674201 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + p = -0.785674201 + -0.5555555555 p = -0.785674201 + -0.5555555555 Combine like terms: -0.785674201 + -0.5555555555 = -1.3412297565 p = -1.3412297565 Simplifying p = -1.3412297565Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.2301186455, -1.3412297565}
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