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Simplifying (r + -3)(r + 15) = -6 Reorder the terms: (-3 + r)(r + 15) = -6 Reorder the terms: (-3 + r)(15 + r) = -6 Multiply (-3 + r) * (15 + r) (-3(15 + r) + r(15 + r)) = -6 ((15 * -3 + r * -3) + r(15 + r)) = -6 ((-45 + -3r) + r(15 + r)) = -6 (-45 + -3r + (15 * r + r * r)) = -6 (-45 + -3r + (15r + r2)) = -6 Combine like terms: -3r + 15r = 12r (-45 + 12r + r2) = -6 Solving -45 + 12r + r2 = -6 Solving for variable 'r'. Reorder the terms: -45 + 6 + 12r + r2 = -6 + 6 Combine like terms: -45 + 6 = -39 -39 + 12r + r2 = -6 + 6 Combine like terms: -6 + 6 = 0 -39 + 12r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '39' to each side of the equation. -39 + 12r + 39 + r2 = 0 + 39 Reorder the terms: -39 + 39 + 12r + r2 = 0 + 39 Combine like terms: -39 + 39 = 0 0 + 12r + r2 = 0 + 39 12r + r2 = 0 + 39 Combine like terms: 0 + 39 = 39 12r + r2 = 39 The r term is 12r. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12r + 36 + r2 = 39 + 36 Reorder the terms: 36 + 12r + r2 = 39 + 36 Combine like terms: 39 + 36 = 75 36 + 12r + r2 = 75 Factor a perfect square on the left side: (r + 6)(r + 6) = 75 Calculate the square root of the right side: 8.660254038 Break this problem into two subproblems by setting (r + 6) equal to 8.660254038 and -8.660254038.Subproblem 1
r + 6 = 8.660254038 Simplifying r + 6 = 8.660254038 Reorder the terms: 6 + r = 8.660254038 Solving 6 + r = 8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = 8.660254038 + -6 Combine like terms: 6 + -6 = 0 0 + r = 8.660254038 + -6 r = 8.660254038 + -6 Combine like terms: 8.660254038 + -6 = 2.660254038 r = 2.660254038 Simplifying r = 2.660254038Subproblem 2
r + 6 = -8.660254038 Simplifying r + 6 = -8.660254038 Reorder the terms: 6 + r = -8.660254038 Solving 6 + r = -8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = -8.660254038 + -6 Combine like terms: 6 + -6 = 0 0 + r = -8.660254038 + -6 r = -8.660254038 + -6 Combine like terms: -8.660254038 + -6 = -14.660254038 r = -14.660254038 Simplifying r = -14.660254038Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.660254038, -14.660254038}
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