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Simplifying 3(r2 + 6r + 21) = 102 Reorder the terms: 3(21 + 6r + r2) = 102 (21 * 3 + 6r * 3 + r2 * 3) = 102 (63 + 18r + 3r2) = 102 Solving 63 + 18r + 3r2 = 102 Solving for variable 'r'. Reorder the terms: 63 + -102 + 18r + 3r2 = 102 + -102 Combine like terms: 63 + -102 = -39 -39 + 18r + 3r2 = 102 + -102 Combine like terms: 102 + -102 = 0 -39 + 18r + 3r2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-13 + 6r + r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-13 + 6r + r2)' equal to zero and attempt to solve: Simplifying -13 + 6r + r2 = 0 Solving -13 + 6r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '13' to each side of the equation. -13 + 6r + 13 + r2 = 0 + 13 Reorder the terms: -13 + 13 + 6r + r2 = 0 + 13 Combine like terms: -13 + 13 = 0 0 + 6r + r2 = 0 + 13 6r + r2 = 0 + 13 Combine like terms: 0 + 13 = 13 6r + r2 = 13 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 = 13 + 9 Reorder the terms: 9 + 6r + r2 = 13 + 9 Combine like terms: 13 + 9 = 22 9 + 6r + r2 = 22 Factor a perfect square on the left side: (r + 3)(r + 3) = 22 Calculate the square root of the right side: 4.69041576 Break this problem into two subproblems by setting (r + 3) equal to 4.69041576 and -4.69041576.Subproblem 1
r + 3 = 4.69041576 Simplifying r + 3 = 4.69041576 Reorder the terms: 3 + r = 4.69041576 Solving 3 + r = 4.69041576 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 = 4.69041576 + -3 Combine like terms: 3 + -3 = 0 0 + r = 4.69041576 + -3 r = 4.69041576 + -3 Combine like terms: 4.69041576 + -3 = 1.69041576 r = 1.69041576 Simplifying r = 1.69041576Subproblem 2
r + 3 = -4.69041576 Simplifying r + 3 = -4.69041576 Reorder the terms: 3 + r = -4.69041576 Solving 3 + r = -4.69041576 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 = -4.69041576 + -3 Combine like terms: 3 + -3 = 0 0 + r = -4.69041576 + -3 r = -4.69041576 + -3 Combine like terms: -4.69041576 + -3 = -7.69041576 r = -7.69041576 Simplifying r = -7.69041576Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.69041576, -7.69041576}Solution
r = {1.69041576, -7.69041576}
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