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Simplifying 0 = 11r2 + -48r + -120 Reorder the terms: 0 = -120 + -48r + 11r2 Solving 0 = -120 + -48r + 11r2 Solving for variable 'r'. Combine like terms: 0 + 120 = 120 120 + 48r + -11r2 = -120 + -48r + 11r2 + 120 + 48r + -11r2 Reorder the terms: 120 + 48r + -11r2 = -120 + 120 + -48r + 48r + 11r2 + -11r2 Combine like terms: -120 + 120 = 0 120 + 48r + -11r2 = 0 + -48r + 48r + 11r2 + -11r2 120 + 48r + -11r2 = -48r + 48r + 11r2 + -11r2 Combine like terms: -48r + 48r = 0 120 + 48r + -11r2 = 0 + 11r2 + -11r2 120 + 48r + -11r2 = 11r2 + -11r2 Combine like terms: 11r2 + -11r2 = 0 120 + 48r + -11r2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -10.90909091 + -4.363636364r + r2 = 0 Move the constant term to the right: Add '10.90909091' to each side of the equation. -10.90909091 + -4.363636364r + 10.90909091 + r2 = 0 + 10.90909091 Reorder the terms: -10.90909091 + 10.90909091 + -4.363636364r + r2 = 0 + 10.90909091 Combine like terms: -10.90909091 + 10.90909091 = 0.00000000 0.00000000 + -4.363636364r + r2 = 0 + 10.90909091 -4.363636364r + r2 = 0 + 10.90909091 Combine like terms: 0 + 10.90909091 = 10.90909091 -4.363636364r + r2 = 10.90909091 The r term is -4.363636364r. Take half its coefficient (-2.181818182). Square it (4.760330579) and add it to both sides. Add '4.760330579' to each side of the equation. -4.363636364r + 4.760330579 + r2 = 10.90909091 + 4.760330579 Reorder the terms: 4.760330579 + -4.363636364r + r2 = 10.90909091 + 4.760330579 Combine like terms: 10.90909091 + 4.760330579 = 15.669421489 4.760330579 + -4.363636364r + r2 = 15.669421489 Factor a perfect square on the left side: (r + -2.181818182)(r + -2.181818182) = 15.669421489 Calculate the square root of the right side: 3.958462011 Break this problem into two subproblems by setting (r + -2.181818182) equal to 3.958462011 and -3.958462011.Subproblem 1
r + -2.181818182 = 3.958462011 Simplifying r + -2.181818182 = 3.958462011 Reorder the terms: -2.181818182 + r = 3.958462011 Solving -2.181818182 + r = 3.958462011 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.181818182' to each side of the equation. -2.181818182 + 2.181818182 + r = 3.958462011 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + r = 3.958462011 + 2.181818182 r = 3.958462011 + 2.181818182 Combine like terms: 3.958462011 + 2.181818182 = 6.140280193 r = 6.140280193 Simplifying r = 6.140280193Subproblem 2
r + -2.181818182 = -3.958462011 Simplifying r + -2.181818182 = -3.958462011 Reorder the terms: -2.181818182 + r = -3.958462011 Solving -2.181818182 + r = -3.958462011 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.181818182' to each side of the equation. -2.181818182 + 2.181818182 + r = -3.958462011 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + r = -3.958462011 + 2.181818182 r = -3.958462011 + 2.181818182 Combine like terms: -3.958462011 + 2.181818182 = -1.776643829 r = -1.776643829 Simplifying r = -1.776643829Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.140280193, -1.776643829}
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