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Simplifying (r + -2)(r + 6) = 29 Reorder the terms: (-2 + r)(r + 6) = 29 Reorder the terms: (-2 + r)(6 + r) = 29 Multiply (-2 + r) * (6 + r) (-2(6 + r) + r(6 + r)) = 29 ((6 * -2 + r * -2) + r(6 + r)) = 29 ((-12 + -2r) + r(6 + r)) = 29 (-12 + -2r + (6 * r + r * r)) = 29 (-12 + -2r + (6r + r2)) = 29 Combine like terms: -2r + 6r = 4r (-12 + 4r + r2) = 29 Solving -12 + 4r + r2 = 29 Solving for variable 'r'. Reorder the terms: -12 + -29 + 4r + r2 = 29 + -29 Combine like terms: -12 + -29 = -41 -41 + 4r + r2 = 29 + -29 Combine like terms: 29 + -29 = 0 -41 + 4r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '41' to each side of the equation. -41 + 4r + 41 + r2 = 0 + 41 Reorder the terms: -41 + 41 + 4r + r2 = 0 + 41 Combine like terms: -41 + 41 = 0 0 + 4r + r2 = 0 + 41 4r + r2 = 0 + 41 Combine like terms: 0 + 41 = 41 4r + r2 = 41 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 = 41 + 4 Reorder the terms: 4 + 4r + r2 = 41 + 4 Combine like terms: 41 + 4 = 45 4 + 4r + r2 = 45 Factor a perfect square on the left side: (r + 2)(r + 2) = 45 Calculate the square root of the right side: 6.708203933 Break this problem into two subproblems by setting (r + 2) equal to 6.708203933 and -6.708203933.Subproblem 1
r + 2 = 6.708203933 Simplifying r + 2 = 6.708203933 Reorder the terms: 2 + r = 6.708203933 Solving 2 + r = 6.708203933 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 = 6.708203933 + -2 Combine like terms: 2 + -2 = 0 0 + r = 6.708203933 + -2 r = 6.708203933 + -2 Combine like terms: 6.708203933 + -2 = 4.708203933 r = 4.708203933 Simplifying r = 4.708203933Subproblem 2
r + 2 = -6.708203933 Simplifying r + 2 = -6.708203933 Reorder the terms: 2 + r = -6.708203933 Solving 2 + r = -6.708203933 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 = -6.708203933 + -2 Combine like terms: 2 + -2 = 0 0 + r = -6.708203933 + -2 r = -6.708203933 + -2 Combine like terms: -6.708203933 + -2 = -8.708203933 r = -8.708203933 Simplifying r = -8.708203933Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.708203933, -8.708203933}
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