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Simplifying 49.5 = r(15 + -1r) 49.5 = (15 * r + -1r * r) 49.5 = (15r + -1r2) Solving 49.5 = 15r + -1r2 Solving for variable 'r'. Reorder the terms: 49.5 + -15r + r2 = 15r + -15r + -1r2 + r2 Combine like terms: 15r + -15r = 0 49.5 + -15r + r2 = 0 + -1r2 + r2 49.5 + -15r + r2 = -1r2 + r2 Combine like terms: -1r2 + r2 = 0 49.5 + -15r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '-49.5' to each side of the equation. 49.5 + -15r + -49.5 + r2 = 0 + -49.5 Reorder the terms: 49.5 + -49.5 + -15r + r2 = 0 + -49.5 Combine like terms: 49.5 + -49.5 = 0.0 0.0 + -15r + r2 = 0 + -49.5 -15r + r2 = 0 + -49.5 Combine like terms: 0 + -49.5 = -49.5 -15r + r2 = -49.5 The r term is -15r. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15r + 56.25 + r2 = -49.5 + 56.25 Reorder the terms: 56.25 + -15r + r2 = -49.5 + 56.25 Combine like terms: -49.5 + 56.25 = 6.75 56.25 + -15r + r2 = 6.75 Factor a perfect square on the left side: (r + -7.5)(r + -7.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (r + -7.5) equal to 2.598076211 and -2.598076211.Subproblem 1
r + -7.5 = 2.598076211 Simplifying r + -7.5 = 2.598076211 Reorder the terms: -7.5 + r = 2.598076211 Solving -7.5 + r = 2.598076211 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + r = 2.598076211 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + r = 2.598076211 + 7.5 r = 2.598076211 + 7.5 Combine like terms: 2.598076211 + 7.5 = 10.098076211 r = 10.098076211 Simplifying r = 10.098076211Subproblem 2
r + -7.5 = -2.598076211 Simplifying r + -7.5 = -2.598076211 Reorder the terms: -7.5 + r = -2.598076211 Solving -7.5 + r = -2.598076211 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + r = -2.598076211 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + r = -2.598076211 + 7.5 r = -2.598076211 + 7.5 Combine like terms: -2.598076211 + 7.5 = 4.901923789 r = 4.901923789 Simplifying r = 4.901923789Solution
The solution to the problem is based on the solutions from the subproblems. r = {10.098076211, 4.901923789}
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