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Simplifying (r + -2)(r + 12) = 31 Reorder the terms: (-2 + r)(r + 12) = 31 Reorder the terms: (-2 + r)(12 + r) = 31 Multiply (-2 + r) * (12 + r) (-2(12 + r) + r(12 + r)) = 31 ((12 * -2 + r * -2) + r(12 + r)) = 31 ((-24 + -2r) + r(12 + r)) = 31 (-24 + -2r + (12 * r + r * r)) = 31 (-24 + -2r + (12r + r2)) = 31 Combine like terms: -2r + 12r = 10r (-24 + 10r + r2) = 31 Solving -24 + 10r + r2 = 31 Solving for variable 'r'. Reorder the terms: -24 + -31 + 10r + r2 = 31 + -31 Combine like terms: -24 + -31 = -55 -55 + 10r + r2 = 31 + -31 Combine like terms: 31 + -31 = 0 -55 + 10r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '55' to each side of the equation. -55 + 10r + 55 + r2 = 0 + 55 Reorder the terms: -55 + 55 + 10r + r2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + 10r + r2 = 0 + 55 10r + r2 = 0 + 55 Combine like terms: 0 + 55 = 55 10r + r2 = 55 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 = 55 + 25 Reorder the terms: 25 + 10r + r2 = 55 + 25 Combine like terms: 55 + 25 = 80 25 + 10r + r2 = 80 Factor a perfect square on the left side: (r + 5)(r + 5) = 80 Calculate the square root of the right side: 8.94427191 Break this problem into two subproblems by setting (r + 5) equal to 8.94427191 and -8.94427191.Subproblem 1
r + 5 = 8.94427191 Simplifying r + 5 = 8.94427191 Reorder the terms: 5 + r = 8.94427191 Solving 5 + r = 8.94427191 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 = 8.94427191 + -5 Combine like terms: 5 + -5 = 0 0 + r = 8.94427191 + -5 r = 8.94427191 + -5 Combine like terms: 8.94427191 + -5 = 3.94427191 r = 3.94427191 Simplifying r = 3.94427191Subproblem 2
r + 5 = -8.94427191 Simplifying r + 5 = -8.94427191 Reorder the terms: 5 + r = -8.94427191 Solving 5 + r = -8.94427191 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 = -8.94427191 + -5 Combine like terms: 5 + -5 = 0 0 + r = -8.94427191 + -5 r = -8.94427191 + -5 Combine like terms: -8.94427191 + -5 = -13.94427191 r = -13.94427191 Simplifying r = -13.94427191Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.94427191, -13.94427191}
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