0=6.28r^2+253.712r-3668.838

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Solution for 0=6.28r^2+253.712r-3668.838 equation:


Simplifying
0 = 6.28r2 + 253.712r + -3668.838

Reorder the terms:
0 = -3668.838 + 253.712r + 6.28r2

Solving
0 = -3668.838 + 253.712r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 3668.838 = 3668.838
3668.838 + -253.712r + -6.28r2 = -3668.838 + 253.712r + 6.28r2 + 3668.838 + -253.712r + -6.28r2

Reorder the terms:
3668.838 + -253.712r + -6.28r2 = -3668.838 + 3668.838 + 253.712r + -253.712r + 6.28r2 + -6.28r2

Combine like terms: -3668.838 + 3668.838 = 0.000
3668.838 + -253.712r + -6.28r2 = 0.000 + 253.712r + -253.712r + 6.28r2 + -6.28r2
3668.838 + -253.712r + -6.28r2 = 253.712r + -253.712r + 6.28r2 + -6.28r2

Combine like terms: 253.712r + -253.712r = 0.000
3668.838 + -253.712r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
3668.838 + -253.712r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
3668.838 + -253.712r + -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'.
-584.2098726 + 40.4r + r2 = 0

Move the constant term to the right:

Add '584.2098726' to each side of the equation.
-584.2098726 + 40.4r + 584.2098726 + r2 = 0 + 584.2098726

Reorder the terms:
-584.2098726 + 584.2098726 + 40.4r + r2 = 0 + 584.2098726

Combine like terms: -584.2098726 + 584.2098726 = 0.0000000
0.0000000 + 40.4r + r2 = 0 + 584.2098726
40.4r + r2 = 0 + 584.2098726

Combine like terms: 0 + 584.2098726 = 584.2098726
40.4r + r2 = 584.2098726

The r term is 40.4r.  Take half its coefficient (20.2).
Square it (408.04) and add it to both sides.

Add '408.04' to each side of the equation.
40.4r + 408.04 + r2 = 584.2098726 + 408.04

Reorder the terms:
408.04 + 40.4r + r2 = 584.2098726 + 408.04

Combine like terms: 584.2098726 + 408.04 = 992.2498726
408.04 + 40.4r + r2 = 992.2498726

Factor a perfect square on the left side:
(r + 20.2)(r + 20.2) = 992.2498726

Calculate the square root of the right side: 31.499997978

Break this problem into two subproblems by setting 
(r + 20.2) equal to 31.499997978 and -31.499997978.

Subproblem 1

r + 20.2 = 31.499997978 Simplifying r + 20.2 = 31.499997978 Reorder the terms: 20.2 + r = 31.499997978 Solving 20.2 + r = 31.499997978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-20.2' to each side of the equation. 20.2 + -20.2 + r = 31.499997978 + -20.2 Combine like terms: 20.2 + -20.2 = 0.0 0.0 + r = 31.499997978 + -20.2 r = 31.499997978 + -20.2 Combine like terms: 31.499997978 + -20.2 = 11.299997978 r = 11.299997978 Simplifying r = 11.299997978

Subproblem 2

r + 20.2 = -31.499997978 Simplifying r + 20.2 = -31.499997978 Reorder the terms: 20.2 + r = -31.499997978 Solving 20.2 + r = -31.499997978 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-20.2' to each side of the equation. 20.2 + -20.2 + r = -31.499997978 + -20.2 Combine like terms: 20.2 + -20.2 = 0.0 0.0 + r = -31.499997978 + -20.2 r = -31.499997978 + -20.2 Combine like terms: -31.499997978 + -20.2 = -51.699997978 r = -51.699997978 Simplifying r = -51.699997978

Solution

The solution to the problem is based on the solutions from the subproblems. r = {11.299997978, -51.699997978}

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