0=6.28r^2+98.596r-509.8104

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Solution for 0=6.28r^2+98.596r-509.8104 equation:


Simplifying
0 = 6.28r2 + 98.596r + -509.8104

Reorder the terms:
0 = -509.8104 + 98.596r + 6.28r2

Solving
0 = -509.8104 + 98.596r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 509.8104 = 509.8104
509.8104 + -98.596r + -6.28r2 = -509.8104 + 98.596r + 6.28r2 + 509.8104 + -98.596r + -6.28r2

Reorder the terms:
509.8104 + -98.596r + -6.28r2 = -509.8104 + 509.8104 + 98.596r + -98.596r + 6.28r2 + -6.28r2

Combine like terms: -509.8104 + 509.8104 = 0.0000
509.8104 + -98.596r + -6.28r2 = 0.0000 + 98.596r + -98.596r + 6.28r2 + -6.28r2
509.8104 + -98.596r + -6.28r2 = 98.596r + -98.596r + 6.28r2 + -6.28r2

Combine like terms: 98.596r + -98.596r = 0.000
509.8104 + -98.596r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
509.8104 + -98.596r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
509.8104 + -98.596r + -6.28r2 = 0.00

Begin completing the square.  Divide all terms by
-6.28 the coefficient of the squared term: 

Divide each side by '-6.28'.
-81.18 + 15.7r + r2 = 0

Move the constant term to the right:

Add '81.18' to each side of the equation.
-81.18 + 15.7r + 81.18 + r2 = 0 + 81.18

Reorder the terms:
-81.18 + 81.18 + 15.7r + r2 = 0 + 81.18

Combine like terms: -81.18 + 81.18 = 0.00
0.00 + 15.7r + r2 = 0 + 81.18
15.7r + r2 = 0 + 81.18

Combine like terms: 0 + 81.18 = 81.18
15.7r + r2 = 81.18

The r term is 15.7r.  Take half its coefficient (7.85).
Square it (61.6225) and add it to both sides.

Add '61.6225' to each side of the equation.
15.7r + 61.6225 + r2 = 81.18 + 61.6225

Reorder the terms:
61.6225 + 15.7r + r2 = 81.18 + 61.6225

Combine like terms: 81.18 + 61.6225 = 142.8025
61.6225 + 15.7r + r2 = 142.8025

Factor a perfect square on the left side:
(r + 7.85)(r + 7.85) = 142.8025

Calculate the square root of the right side: 11.95

Break this problem into two subproblems by setting 
(r + 7.85) equal to 11.95 and -11.95.

Subproblem 1

r + 7.85 = 11.95 Simplifying r + 7.85 = 11.95 Reorder the terms: 7.85 + r = 11.95 Solving 7.85 + r = 11.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.85' to each side of the equation. 7.85 + -7.85 + r = 11.95 + -7.85 Combine like terms: 7.85 + -7.85 = 0.00 0.00 + r = 11.95 + -7.85 r = 11.95 + -7.85 Combine like terms: 11.95 + -7.85 = 4.1 r = 4.1 Simplifying r = 4.1

Subproblem 2

r + 7.85 = -11.95 Simplifying r + 7.85 = -11.95 Reorder the terms: 7.85 + r = -11.95 Solving 7.85 + r = -11.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.85' to each side of the equation. 7.85 + -7.85 + r = -11.95 + -7.85 Combine like terms: 7.85 + -7.85 = 0.00 0.00 + r = -11.95 + -7.85 r = -11.95 + -7.85 Combine like terms: -11.95 + -7.85 = -19.8 r = -19.8 Simplifying r = -19.8

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.1, -19.8}

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