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Simplifying (r + 4)(r + 2) = 1 Reorder the terms: (4 + r)(r + 2) = 1 Reorder the terms: (4 + r)(2 + r) = 1 Multiply (4 + r) * (2 + r) (4(2 + r) + r(2 + r)) = 1 ((2 * 4 + r * 4) + r(2 + r)) = 1 ((8 + 4r) + r(2 + r)) = 1 (8 + 4r + (2 * r + r * r)) = 1 (8 + 4r + (2r + r2)) = 1 Combine like terms: 4r + 2r = 6r (8 + 6r + r2) = 1 Solving 8 + 6r + r2 = 1 Solving for variable 'r'. Reorder the terms: 8 + -1 + 6r + r2 = 1 + -1 Combine like terms: 8 + -1 = 7 7 + 6r + r2 = 1 + -1 Combine like terms: 1 + -1 = 0 7 + 6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + 6r + -7 + r2 = 0 + -7 Reorder the terms: 7 + -7 + 6r + r2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 6r + r2 = 0 + -7 6r + r2 = 0 + -7 Combine like terms: 0 + -7 = -7 6r + r2 = -7 The r term is 6r. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6r + 9 + r2 = -7 + 9 Reorder the terms: 9 + 6r + r2 = -7 + 9 Combine like terms: -7 + 9 = 2 9 + 6r + r2 = 2 Factor a perfect square on the left side: (r + 3)(r + 3) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (r + 3) equal to 1.414213562 and -1.414213562.Subproblem 1
r + 3 = 1.414213562 Simplifying r + 3 = 1.414213562 Reorder the terms: 3 + r = 1.414213562 Solving 3 + r = 1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + r = 1.414213562 + -3 r = 1.414213562 + -3 Combine like terms: 1.414213562 + -3 = -1.585786438 r = -1.585786438 Simplifying r = -1.585786438Subproblem 2
r + 3 = -1.414213562 Simplifying r + 3 = -1.414213562 Reorder the terms: 3 + r = -1.414213562 Solving 3 + r = -1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -1.414213562 + -3 Combine like terms: 3 + -3 = 0 0 + r = -1.414213562 + -3 r = -1.414213562 + -3 Combine like terms: -1.414213562 + -3 = -4.414213562 r = -4.414213562 Simplifying r = -4.414213562Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.585786438, -4.414213562}
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