322.816=6.28r^2+90.432r

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Solution for 322.816=6.28r^2+90.432r equation:


Simplifying
322.816 = 6.28r2 + 90.432r

Reorder the terms:
322.816 = 90.432r + 6.28r2

Solving
322.816 = 90.432r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
322.816 + -90.432r + -6.28r2 = 90.432r + -90.432r + 6.28r2 + -6.28r2

Combine like terms: 90.432r + -90.432r = 0.000
322.816 + -90.432r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
322.816 + -90.432r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
322.816 + -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'.
-51.40382166 + 14.4r + r2 = 0

Move the constant term to the right:

Add '51.40382166' to each side of the equation.
-51.40382166 + 14.4r + 51.40382166 + r2 = 0 + 51.40382166

Reorder the terms:
-51.40382166 + 51.40382166 + 14.4r + r2 = 0 + 51.40382166

Combine like terms: -51.40382166 + 51.40382166 = 0.00000000
0.00000000 + 14.4r + r2 = 0 + 51.40382166
14.4r + r2 = 0 + 51.40382166

Combine like terms: 0 + 51.40382166 = 51.40382166
14.4r + r2 = 51.40382166

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 = 51.40382166 + 51.84

Reorder the terms:
51.84 + 14.4r + r2 = 51.40382166 + 51.84

Combine like terms: 51.40382166 + 51.84 = 103.24382166
51.84 + 14.4r + r2 = 103.24382166

Factor a perfect square on the left side:
(r + 7.2)(r + 7.2) = 103.24382166

Calculate the square root of the right side: 10.160896696

Break this problem into two subproblems by setting 
(r + 7.2) equal to 10.160896696 and -10.160896696.

Subproblem 1

r + 7.2 = 10.160896696 Simplifying r + 7.2 = 10.160896696 Reorder the terms: 7.2 + r = 10.160896696 Solving 7.2 + r = 10.160896696 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 = 10.160896696 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = 10.160896696 + -7.2 r = 10.160896696 + -7.2 Combine like terms: 10.160896696 + -7.2 = 2.960896696 r = 2.960896696 Simplifying r = 2.960896696

Subproblem 2

r + 7.2 = -10.160896696 Simplifying r + 7.2 = -10.160896696 Reorder the terms: 7.2 + r = -10.160896696 Solving 7.2 + r = -10.160896696 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 = -10.160896696 + -7.2 Combine like terms: 7.2 + -7.2 = 0.0 0.0 + r = -10.160896696 + -7.2 r = -10.160896696 + -7.2 Combine like terms: -10.160896696 + -7.2 = -17.360896696 r = -17.360896696 Simplifying r = -17.360896696

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.960896696, -17.360896696}

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