188.9024=6.28r^2+20.069r

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Solution for 188.9024=6.28r^2+20.069r equation:


Simplifying
188.9024 = 6.28r2 + 20.069r

Reorder the terms:
188.9024 = 20.069r + 6.28r2

Solving
188.9024 = 20.069r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
188.9024 + -20.069r + -6.28r2 = 20.069r + -20.069r + 6.28r2 + -6.28r2

Combine like terms: 20.069r + -20.069r = 0.000
188.9024 + -20.069r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
188.9024 + -20.069r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
188.9024 + -20.069r + -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'.
-30.08 + 3.195700637r + r2 = 0

Move the constant term to the right:

Add '30.08' to each side of the equation.
-30.08 + 3.195700637r + 30.08 + r2 = 0 + 30.08

Reorder the terms:
-30.08 + 30.08 + 3.195700637r + r2 = 0 + 30.08

Combine like terms: -30.08 + 30.08 = 0.00
0.00 + 3.195700637r + r2 = 0 + 30.08
3.195700637r + r2 = 0 + 30.08

Combine like terms: 0 + 30.08 = 30.08
3.195700637r + r2 = 30.08

The r term is 3.195700637r.  Take half its coefficient (1.597850319).
Square it (2.553125642) and add it to both sides.

Add '2.553125642' to each side of the equation.
3.195700637r + 2.553125642 + r2 = 30.08 + 2.553125642

Reorder the terms:
2.553125642 + 3.195700637r + r2 = 30.08 + 2.553125642

Combine like terms: 30.08 + 2.553125642 = 32.633125642
2.553125642 + 3.195700637r + r2 = 32.633125642

Factor a perfect square on the left side:
(r + 1.597850319)(r + 1.597850319) = 32.633125642

Calculate the square root of the right side: 5.712541084

Break this problem into two subproblems by setting 
(r + 1.597850319) equal to 5.712541084 and -5.712541084.

Subproblem 1

r + 1.597850319 = 5.712541084 Simplifying r + 1.597850319 = 5.712541084 Reorder the terms: 1.597850319 + r = 5.712541084 Solving 1.597850319 + r = 5.712541084 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.597850319' to each side of the equation. 1.597850319 + -1.597850319 + r = 5.712541084 + -1.597850319 Combine like terms: 1.597850319 + -1.597850319 = 0.000000000 0.000000000 + r = 5.712541084 + -1.597850319 r = 5.712541084 + -1.597850319 Combine like terms: 5.712541084 + -1.597850319 = 4.114690765 r = 4.114690765 Simplifying r = 4.114690765

Subproblem 2

r + 1.597850319 = -5.712541084 Simplifying r + 1.597850319 = -5.712541084 Reorder the terms: 1.597850319 + r = -5.712541084 Solving 1.597850319 + r = -5.712541084 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.597850319' to each side of the equation. 1.597850319 + -1.597850319 + r = -5.712541084 + -1.597850319 Combine like terms: 1.597850319 + -1.597850319 = 0.000000000 0.000000000 + r = -5.712541084 + -1.597850319 r = -5.712541084 + -1.597850319 Combine like terms: -5.712541084 + -1.597850319 = -7.310391403 r = -7.310391403 Simplifying r = -7.310391403

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.114690765, -7.310391403}

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