0=6.28r^2+589.692r-3040.2736

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Solution for 0=6.28r^2+589.692r-3040.2736 equation:


Simplifying
0 = 6.28r2 + 589.692r + -3040.2736

Reorder the terms:
0 = -3040.2736 + 589.692r + 6.28r2

Solving
0 = -3040.2736 + 589.692r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 3040.2736 = 3040.2736
3040.2736 + -589.692r + -6.28r2 = -3040.2736 + 589.692r + 6.28r2 + 3040.2736 + -589.692r + -6.28r2

Reorder the terms:
3040.2736 + -589.692r + -6.28r2 = -3040.2736 + 3040.2736 + 589.692r + -589.692r + 6.28r2 + -6.28r2

Combine like terms: -3040.2736 + 3040.2736 = 0.0000
3040.2736 + -589.692r + -6.28r2 = 0.0000 + 589.692r + -589.692r + 6.28r2 + -6.28r2
3040.2736 + -589.692r + -6.28r2 = 589.692r + -589.692r + 6.28r2 + -6.28r2

Combine like terms: 589.692r + -589.692r = 0.000
3040.2736 + -589.692r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
3040.2736 + -589.692r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
3040.2736 + -589.692r + -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'.
-484.12 + 93.9r + r2 = 0

Move the constant term to the right:

Add '484.12' to each side of the equation.
-484.12 + 93.9r + 484.12 + r2 = 0 + 484.12

Reorder the terms:
-484.12 + 484.12 + 93.9r + r2 = 0 + 484.12

Combine like terms: -484.12 + 484.12 = 0.00
0.00 + 93.9r + r2 = 0 + 484.12
93.9r + r2 = 0 + 484.12

Combine like terms: 0 + 484.12 = 484.12
93.9r + r2 = 484.12

The r term is 93.9r.  Take half its coefficient (46.95).
Square it (2204.3025) and add it to both sides.

Add '2204.3025' to each side of the equation.
93.9r + 2204.3025 + r2 = 484.12 + 2204.3025

Reorder the terms:
2204.3025 + 93.9r + r2 = 484.12 + 2204.3025

Combine like terms: 484.12 + 2204.3025 = 2688.4225
2204.3025 + 93.9r + r2 = 2688.4225

Factor a perfect square on the left side:
(r + 46.95)(r + 46.95) = 2688.4225

Calculate the square root of the right side: 51.85

Break this problem into two subproblems by setting 
(r + 46.95) equal to 51.85 and -51.85.

Subproblem 1

r + 46.95 = 51.85 Simplifying r + 46.95 = 51.85 Reorder the terms: 46.95 + r = 51.85 Solving 46.95 + r = 51.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = 51.85 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = 51.85 + -46.95 r = 51.85 + -46.95 Combine like terms: 51.85 + -46.95 = 4.9 r = 4.9 Simplifying r = 4.9

Subproblem 2

r + 46.95 = -51.85 Simplifying r + 46.95 = -51.85 Reorder the terms: 46.95 + r = -51.85 Solving 46.95 + r = -51.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = -51.85 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = -51.85 + -46.95 r = -51.85 + -46.95 Combine like terms: -51.85 + -46.95 = -98.8 r = -98.8 Simplifying r = -98.8

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.9, -98.8}

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