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Simplifying (r + 1)(r + 1) = 14 Reorder the terms: (1 + r)(r + 1) = 14 Reorder the terms: (1 + r)(1 + r) = 14 Multiply (1 + r) * (1 + r) (1(1 + r) + r(1 + r)) = 14 ((1 * 1 + r * 1) + r(1 + r)) = 14 ((1 + 1r) + r(1 + r)) = 14 (1 + 1r + (1 * r + r * r)) = 14 (1 + 1r + (1r + r2)) = 14 Combine like terms: 1r + 1r = 2r (1 + 2r + r2) = 14 Solving 1 + 2r + r2 = 14 Solving for variable 'r'. Reorder the terms: 1 + -14 + 2r + r2 = 14 + -14 Combine like terms: 1 + -14 = -13 -13 + 2r + r2 = 14 + -14 Combine like terms: 14 + -14 = 0 -13 + 2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '13' to each side of the equation. -13 + 2r + 13 + r2 = 0 + 13 Reorder the terms: -13 + 13 + 2r + r2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + 2r + r2 = 0 + 13 2r + r2 = 0 + 13 Combine like terms: 0 + 13 = 13 2r + r2 = 13 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 13 + 1 Reorder the terms: 1 + 2r + r2 = 13 + 1 Combine like terms: 13 + 1 = 14 1 + 2r + r2 = 14 Factor a perfect square on the left side: (r + 1)(r + 1) = 14 Calculate the square root of the right side: 3.741657387 Break this problem into two subproblems by setting (r + 1) equal to 3.741657387 and -3.741657387.Subproblem 1
r + 1 = 3.741657387 Simplifying r + 1 = 3.741657387 Reorder the terms: 1 + r = 3.741657387 Solving 1 + r = 3.741657387 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 3.741657387 + -1 Combine like terms: 1 + -1 = 0 0 + r = 3.741657387 + -1 r = 3.741657387 + -1 Combine like terms: 3.741657387 + -1 = 2.741657387 r = 2.741657387 Simplifying r = 2.741657387Subproblem 2
r + 1 = -3.741657387 Simplifying r + 1 = -3.741657387 Reorder the terms: 1 + r = -3.741657387 Solving 1 + r = -3.741657387 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -3.741657387 + -1 Combine like terms: 1 + -1 = 0 0 + r = -3.741657387 + -1 r = -3.741657387 + -1 Combine like terms: -3.741657387 + -1 = -4.741657387 r = -4.741657387 Simplifying r = -4.741657387Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.741657387, -4.741657387}
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