3r^2-122r+500=0

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Solution for 3r^2-122r+500=0 equation:


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 = 36.042495571

Subproblem 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 = 4.624171109

Solution

The solution to the problem is based on the solutions from the subproblems. r = {36.042495571, 4.624171109}

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