3r^2-14r+12=0

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Solution for 3r^2-14r+12=0 equation:


Simplifying
3r2 + -14r + 12 = 0

Reorder the terms:
12 + -14r + 3r2 = 0

Solving
12 + -14r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
4 + -4.666666667r + r2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -4.666666667r + -4 + r2 = 0 + -4

Reorder the terms:
4 + -4 + -4.666666667r + r2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -4.666666667r + r2 = 0 + -4
-4.666666667r + r2 = 0 + -4

Combine like terms: 0 + -4 = -4
-4.666666667r + r2 = -4

The r term is -4.666666667r.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667r + 5.444444448 + r2 = -4 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667r + r2 = -4 + 5.444444448

Combine like terms: -4 + 5.444444448 = 1.444444448
5.444444448 + -4.666666667r + r2 = 1.444444448

Factor a perfect square on the left side:
(r + -2.333333334)(r + -2.333333334) = 1.444444448

Calculate the square root of the right side: 1.201850427

Break this problem into two subproblems by setting 
(r + -2.333333334) equal to 1.201850427 and -1.201850427.


Subproblem 1
r + -2.333333334 = 1.201850427

Simplifying
r + -2.333333334 = 1.201850427

Reorder the terms:
-2.333333334 + r = 1.201850427

Solving
-2.333333334 + r = 1.201850427

Solving for variable 'r'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + r = 1.201850427 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + r = 1.201850427 + 2.333333334
r = 1.201850427 + 2.333333334

Combine like terms: 1.201850427 + 2.333333334 = 3.535183761
r = 3.535183761

Simplifying
r = 3.535183761


Subproblem 2
r + -2.333333334 = -1.201850427

Simplifying
r + -2.333333334 = -1.201850427

Reorder the terms:
-2.333333334 + r = -1.201850427

Solving
-2.333333334 + r = -1.201850427

Solving for variable 'r'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + r = -1.201850427 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + r = -1.201850427 + 2.333333334
r = -1.201850427 + 2.333333334

Combine like terms: -1.201850427 + 2.333333334 = 1.131482907
r = 1.131482907

Simplifying
r = 1.131482907


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.535183761, 1.131482907}

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