0=3r^2-8r+1

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Solution for 0=3r^2-8r+1 equation:


Simplifying
0 = 3r2 + -8r + 1

Reorder the terms:
0 = 1 + -8r + 3r2

Solving
0 = 1 + -8r + 3r2

Solving for variable 'r'.

Combine like terms: 0 + -1 = -1
-1 + 8r + -3r2 = 1 + -8r + 3r2 + -1 + 8r + -3r2

Reorder the terms:
-1 + 8r + -3r2 = 1 + -1 + -8r + 8r + 3r2 + -3r2

Combine like terms: 1 + -1 = 0
-1 + 8r + -3r2 = 0 + -8r + 8r + 3r2 + -3r2
-1 + 8r + -3r2 = -8r + 8r + 3r2 + -3r2

Combine like terms: -8r + 8r = 0
-1 + 8r + -3r2 = 0 + 3r2 + -3r2
-1 + 8r + -3r2 = 3r2 + -3r2

Combine like terms: 3r2 + -3r2 = 0
-1 + 8r + -3r2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
0.3333333333 + -2.666666667r + r2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2.666666667r + -0.3333333333 + r2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2.666666667r + r2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2.666666667r + r2 = 0 + -0.3333333333
-2.666666667r + r2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2.666666667r + r2 = -0.3333333333

The r term is -2.666666667r.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667r + 1.777777780 + r2 = -0.3333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667r + r2 = -0.3333333333 + 1.777777780

Combine like terms: -0.3333333333 + 1.777777780 = 1.4444444467
1.777777780 + -2.666666667r + r2 = 1.4444444467

Factor a perfect square on the left side:
(r + -1.333333334)(r + -1.333333334) = 1.4444444467

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(r + -1.333333334) equal to 1.201850426 and -1.201850426.

Subproblem 1

r + -1.333333334 = 1.201850426 Simplifying r + -1.333333334 = 1.201850426 Reorder the terms: -1.333333334 + r = 1.201850426 Solving -1.333333334 + r = 1.201850426 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = 1.201850426 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = 1.201850426 + 1.333333334 r = 1.201850426 + 1.333333334 Combine like terms: 1.201850426 + 1.333333334 = 2.53518376 r = 2.53518376 Simplifying r = 2.53518376

Subproblem 2

r + -1.333333334 = -1.201850426 Simplifying r + -1.333333334 = -1.201850426 Reorder the terms: -1.333333334 + r = -1.201850426 Solving -1.333333334 + r = -1.201850426 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + r = -1.201850426 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + r = -1.201850426 + 1.333333334 r = -1.201850426 + 1.333333334 Combine like terms: -1.201850426 + 1.333333334 = 0.131482908 r = 0.131482908 Simplifying r = 0.131482908

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.53518376, 0.131482908}

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