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Simplifying 3r2 + -122r + 500 = 0 Reorder the terms: 500 + -122r + 3r2 = 0 Solving 500 + -122r + 3r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 166.6666667 + -40.66666667r + r2 = 0 Move the constant term to the right: Add '-166.6666667' to each side of the equation. 166.6666667 + -40.66666667r + -166.6666667 + r2 = 0 + -166.6666667 Reorder the terms: 166.6666667 + -166.6666667 + -40.66666667r + r2 = 0 + -166.6666667 Combine like terms: 166.6666667 + -166.6666667 = 0.0000000 0.0000000 + -40.66666667r + r2 = 0 + -166.6666667 -40.66666667r + r2 = 0 + -166.6666667 Combine like terms: 0 + -166.6666667 = -166.6666667 -40.66666667r + r2 = -166.6666667 The r term is -40.66666667r. Take half its coefficient (-20.33333334). Square it (413.4444447) and add it to both sides. Add '413.4444447' to each side of the equation. -40.66666667r + 413.4444447 + r2 = -166.6666667 + 413.4444447 Reorder the terms: 413.4444447 + -40.66666667r + r2 = -166.6666667 + 413.4444447 Combine like terms: -166.6666667 + 413.4444447 = 246.777778 413.4444447 + -40.66666667r + r2 = 246.777778 Factor a perfect square on the left side: (r + -20.33333334)(r + -20.33333334) = 246.777778 Calculate the square root of the right side: 15.709162231 Break this problem into two subproblems by setting (r + -20.33333334) equal to 15.709162231 and -15.709162231.Subproblem 1
r + -20.33333334 = 15.709162231 Simplifying r + -20.33333334 = 15.709162231 Reorder the terms: -20.33333334 + r = 15.709162231 Solving -20.33333334 + r = 15.709162231 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '20.33333334' to each side of the equation. -20.33333334 + 20.33333334 + r = 15.709162231 + 20.33333334 Combine like terms: -20.33333334 + 20.33333334 = 0.00000000 0.00000000 + r = 15.709162231 + 20.33333334 r = 15.709162231 + 20.33333334 Combine like terms: 15.709162231 + 20.33333334 = 36.042495571 r = 36.042495571 Simplifying r = 36.042495571Subproblem 2
r + -20.33333334 = -15.709162231 Simplifying r + -20.33333334 = -15.709162231 Reorder the terms: -20.33333334 + r = -15.709162231 Solving -20.33333334 + r = -15.709162231 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '20.33333334' to each side of the equation. -20.33333334 + 20.33333334 + r = -15.709162231 + 20.33333334 Combine like terms: -20.33333334 + 20.33333334 = 0.00000000 0.00000000 + r = -15.709162231 + 20.33333334 r = -15.709162231 + 20.33333334 Combine like terms: -15.709162231 + 20.33333334 = 4.624171109 r = 4.624171109 Simplifying r = 4.624171109Solution
The solution to the problem is based on the solutions from the subproblems. r = {36.042495571, 4.624171109}
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