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Simplifying (r + -2)(r + 6) = 11 Reorder the terms: (-2 + r)(r + 6) = 11 Reorder the terms: (-2 + r)(6 + r) = 11 Multiply (-2 + r) * (6 + r) (-2(6 + r) + r(6 + r)) = 11 ((6 * -2 + r * -2) + r(6 + r)) = 11 ((-12 + -2r) + r(6 + r)) = 11 (-12 + -2r + (6 * r + r * r)) = 11 (-12 + -2r + (6r + r2)) = 11 Combine like terms: -2r + 6r = 4r (-12 + 4r + r2) = 11 Solving -12 + 4r + r2 = 11 Solving for variable 'r'. Reorder the terms: -12 + -11 + 4r + r2 = 11 + -11 Combine like terms: -12 + -11 = -23 -23 + 4r + r2 = 11 + -11 Combine like terms: 11 + -11 = 0 -23 + 4r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 4r + 23 + r2 = 0 + 23 Reorder the terms: -23 + 23 + 4r + r2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 4r + r2 = 0 + 23 4r + r2 = 0 + 23 Combine like terms: 0 + 23 = 23 4r + r2 = 23 The r term is 4r. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4r + 4 + r2 = 23 + 4 Reorder the terms: 4 + 4r + r2 = 23 + 4 Combine like terms: 23 + 4 = 27 4 + 4r + r2 = 27 Factor a perfect square on the left side: (r + 2)(r + 2) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (r + 2) equal to 5.196152423 and -5.196152423.Subproblem 1
r + 2 = 5.196152423 Simplifying r + 2 = 5.196152423 Reorder the terms: 2 + r = 5.196152423 Solving 2 + r = 5.196152423 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = 5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + r = 5.196152423 + -2 r = 5.196152423 + -2 Combine like terms: 5.196152423 + -2 = 3.196152423 r = 3.196152423 Simplifying r = 3.196152423Subproblem 2
r + 2 = -5.196152423 Simplifying r + 2 = -5.196152423 Reorder the terms: 2 + r = -5.196152423 Solving 2 + r = -5.196152423 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + r = -5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + r = -5.196152423 + -2 r = -5.196152423 + -2 Combine like terms: -5.196152423 + -2 = -7.196152423 r = -7.196152423 Simplifying r = -7.196152423Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.196152423, -7.196152423}
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