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Simplifying (r + -4)(r + 12) = 11 Reorder the terms: (-4 + r)(r + 12) = 11 Reorder the terms: (-4 + r)(12 + r) = 11 Multiply (-4 + r) * (12 + r) (-4(12 + r) + r(12 + r)) = 11 ((12 * -4 + r * -4) + r(12 + r)) = 11 ((-48 + -4r) + r(12 + r)) = 11 (-48 + -4r + (12 * r + r * r)) = 11 (-48 + -4r + (12r + r2)) = 11 Combine like terms: -4r + 12r = 8r (-48 + 8r + r2) = 11 Solving -48 + 8r + r2 = 11 Solving for variable 'r'. Reorder the terms: -48 + -11 + 8r + r2 = 11 + -11 Combine like terms: -48 + -11 = -59 -59 + 8r + r2 = 11 + -11 Combine like terms: 11 + -11 = 0 -59 + 8r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '59' to each side of the equation. -59 + 8r + 59 + r2 = 0 + 59 Reorder the terms: -59 + 59 + 8r + r2 = 0 + 59 Combine like terms: -59 + 59 = 0 0 + 8r + r2 = 0 + 59 8r + r2 = 0 + 59 Combine like terms: 0 + 59 = 59 8r + r2 = 59 The r term is 8r. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8r + 16 + r2 = 59 + 16 Reorder the terms: 16 + 8r + r2 = 59 + 16 Combine like terms: 59 + 16 = 75 16 + 8r + r2 = 75 Factor a perfect square on the left side: (r + 4)(r + 4) = 75 Calculate the square root of the right side: 8.660254038 Break this problem into two subproblems by setting (r + 4) equal to 8.660254038 and -8.660254038.Subproblem 1
r + 4 = 8.660254038 Simplifying r + 4 = 8.660254038 Reorder the terms: 4 + r = 8.660254038 Solving 4 + r = 8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + r = 8.660254038 + -4 Combine like terms: 4 + -4 = 0 0 + r = 8.660254038 + -4 r = 8.660254038 + -4 Combine like terms: 8.660254038 + -4 = 4.660254038 r = 4.660254038 Simplifying r = 4.660254038Subproblem 2
r + 4 = -8.660254038 Simplifying r + 4 = -8.660254038 Reorder the terms: 4 + r = -8.660254038 Solving 4 + r = -8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + r = -8.660254038 + -4 Combine like terms: 4 + -4 = 0 0 + r = -8.660254038 + -4 r = -8.660254038 + -4 Combine like terms: -8.660254038 + -4 = -12.660254038 r = -12.660254038 Simplifying r = -12.660254038Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.660254038, -12.660254038}
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