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Simplifying (r + -2)(r + 12) = -1 Reorder the terms: (-2 + r)(r + 12) = -1 Reorder the terms: (-2 + r)(12 + r) = -1 Multiply (-2 + r) * (12 + r) (-2(12 + r) + r(12 + r)) = -1 ((12 * -2 + r * -2) + r(12 + r)) = -1 ((-24 + -2r) + r(12 + r)) = -1 (-24 + -2r + (12 * r + r * r)) = -1 (-24 + -2r + (12r + r2)) = -1 Combine like terms: -2r + 12r = 10r (-24 + 10r + r2) = -1 Solving -24 + 10r + r2 = -1 Solving for variable 'r'. Reorder the terms: -24 + 1 + 10r + r2 = -1 + 1 Combine like terms: -24 + 1 = -23 -23 + 10r + r2 = -1 + 1 Combine like terms: -1 + 1 = 0 -23 + 10r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 10r + 23 + r2 = 0 + 23 Reorder the terms: -23 + 23 + 10r + r2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 10r + r2 = 0 + 23 10r + r2 = 0 + 23 Combine like terms: 0 + 23 = 23 10r + r2 = 23 The r term is 10r. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10r + 25 + r2 = 23 + 25 Reorder the terms: 25 + 10r + r2 = 23 + 25 Combine like terms: 23 + 25 = 48 25 + 10r + r2 = 48 Factor a perfect square on the left side: (r + 5)(r + 5) = 48 Calculate the square root of the right side: 6.92820323 Break this problem into two subproblems by setting (r + 5) equal to 6.92820323 and -6.92820323.Subproblem 1
r + 5 = 6.92820323 Simplifying r + 5 = 6.92820323 Reorder the terms: 5 + r = 6.92820323 Solving 5 + r = 6.92820323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = 6.92820323 + -5 Combine like terms: 5 + -5 = 0 0 + r = 6.92820323 + -5 r = 6.92820323 + -5 Combine like terms: 6.92820323 + -5 = 1.92820323 r = 1.92820323 Simplifying r = 1.92820323Subproblem 2
r + 5 = -6.92820323 Simplifying r + 5 = -6.92820323 Reorder the terms: 5 + r = -6.92820323 Solving 5 + r = -6.92820323 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = -6.92820323 + -5 Combine like terms: 5 + -5 = 0 0 + r = -6.92820323 + -5 r = -6.92820323 + -5 Combine like terms: -6.92820323 + -5 = -11.92820323 r = -11.92820323 Simplifying r = -11.92820323Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.92820323, -11.92820323}
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