0=6.28r^2+90.432r-327.816

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Solution for 0=6.28r^2+90.432r-327.816 equation:


Simplifying
0 = 6.28r2 + 90.432r + -327.816

Reorder the terms:
0 = -327.816 + 90.432r + 6.28r2

Solving
0 = -327.816 + 90.432r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 327.816 = 327.816
327.816 + -90.432r + -6.28r2 = -327.816 + 90.432r + 6.28r2 + 327.816 + -90.432r + -6.28r2

Reorder the terms:
327.816 + -90.432r + -6.28r2 = -327.816 + 327.816 + 90.432r + -90.432r + 6.28r2 + -6.28r2

Combine like terms: -327.816 + 327.816 = 0.000
327.816 + -90.432r + -6.28r2 = 0.000 + 90.432r + -90.432r + 6.28r2 + -6.28r2
327.816 + -90.432r + -6.28r2 = 90.432r + -90.432r + 6.28r2 + -6.28r2

Combine like terms: 90.432r + -90.432r = 0.000
327.816 + -90.432r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
327.816 + -90.432r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
327.816 + -90.432r + -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'.
-52.2 + 14.4r + r2 = 0

Move the constant term to the right:

Add '52.2' to each side of the equation.
-52.2 + 14.4r + 52.2 + r2 = 0 + 52.2

Reorder the terms:
-52.2 + 52.2 + 14.4r + r2 = 0 + 52.2

Combine like terms: -52.2 + 52.2 = 0.0
0.0 + 14.4r + r2 = 0 + 52.2
14.4r + r2 = 0 + 52.2

Combine like terms: 0 + 52.2 = 52.2
14.4r + r2 = 52.2

The r term is 14.4r.  Take half its coefficient (7.2).
Square it (51.84) and add it to both sides.

Add '51.84' to each side of the equation.
14.4r + 51.84 + r2 = 52.2 + 51.84

Reorder the terms:
51.84 + 14.4r + r2 = 52.2 + 51.84

Combine like terms: 52.2 + 51.84 = 104.04
51.84 + 14.4r + r2 = 104.04

Factor a perfect square on the left side:
(r + 7.2)(r + 7.2) = 104.04

Calculate the square root of the right side: 10.2

Break this problem into two subproblems by setting 
(r + 7.2) equal to 10.2 and -10.2.

Subproblem 1

r + 7.2 = 10.2 Simplifying r + 7.2 = 10.2 Reorder the terms: 7.2 + r = 10.2 Solving 7.2 + r = 10.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.2' to each side of the equation. 7.2 + -7.2 + r = 10.2 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = 10.2 + -7.2 r = 10.2 + -7.2 Combine like terms: 10.2 + -7.2 = 3 r = 3 Simplifying r = 3

Subproblem 2

r + 7.2 = -10.2 Simplifying r + 7.2 = -10.2 Reorder the terms: 7.2 + r = -10.2 Solving 7.2 + r = -10.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.2' to each side of the equation. 7.2 + -7.2 + r = -10.2 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = -10.2 + -7.2 r = -10.2 + -7.2 Combine like terms: -10.2 + -7.2 = -17.4 r = -17.4 Simplifying r = -17.4

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3, -17.4}

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