25.748=6.28r^2+6.28*r*3.1

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Solution for 25.748=6.28r^2+6.28*r*3.1 equation: 


Simplifying
25.748 = 6.28r2 + 6.28r * 3.1

Reorder the terms for easier multiplication:
25.748 = 6.28r2 + 6.28 * 3.1r

Multiply 6.28 * 3.1
25.748 = 6.28r2 + 19.468r

Reorder the terms:
25.748 = 19.468r + 6.28r2

Solving
25.748 = 19.468r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
25.748 + -19.468r + -6.28r2 = 19.468r + -19.468r + 6.28r2 + -6.28r2

Combine like terms: 19.468r + -19.468r = 0.000
25.748 + -19.468r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
25.748 + -19.468r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
25.748 + -19.468r + -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'.
-4.1 + 3.1r + r2 = 0

Move the constant term to the right:

Add '4.1' to each side of the equation.
-4.1 + 3.1r + 4.1 + r2 = 0 + 4.1

Reorder the terms:
-4.1 + 4.1 + 3.1r + r2 = 0 + 4.1

Combine like terms: -4.1 + 4.1 = 0.0
0.0 + 3.1r + r2 = 0 + 4.1
3.1r + r2 = 0 + 4.1

Combine like terms: 0 + 4.1 = 4.1
3.1r + r2 = 4.1

The r term is 3.1r.  Take half its coefficient (1.55).
Square it (2.4025) and add it to both sides.

Add '2.4025' to each side of the equation.
3.1r + 2.4025 + r2 = 4.1 + 2.4025

Reorder the terms:
2.4025 + 3.1r + r2 = 4.1 + 2.4025

Combine like terms: 4.1 + 2.4025 = 6.5025
2.4025 + 3.1r + r2 = 6.5025

Factor a perfect square on the left side:
(r + 1.55)(r + 1.55) = 6.5025

Calculate the square root of the right side: 2.55

Break this problem into two subproblems by setting 
(r + 1.55) equal to 2.55 and -2.55.


Subproblem 1
r + 1.55 = 2.55

Simplifying
r + 1.55 = 2.55

Reorder the terms:
1.55 + r = 2.55

Solving
1.55 + r = 2.55

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-1.55' to each side of the equation.
1.55 + -1.55 + r = 2.55 + -1.55

Combine like terms: 1.55 + -1.55 = 0.00
0.00 + r = 2.55 + -1.55
r = 2.55 + -1.55

Combine like terms: 2.55 + -1.55 = 1
r = 1

Simplifying
r = 1


Subproblem 2
r + 1.55 = -2.55

Simplifying
r + 1.55 = -2.55

Reorder the terms:
1.55 + r = -2.55

Solving
1.55 + r = -2.55

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-1.55' to each side of the equation.
1.55 + -1.55 + r = -2.55 + -1.55

Combine like terms: 1.55 + -1.55 = 0.00
0.00 + r = -2.55 + -1.55
r = -2.55 + -1.55

Combine like terms: -2.55 + -1.55 = -4.1
r = -4.1

Simplifying
r = -4.1


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1, -4.1}

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