r+3r(r-6)=1

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Solution for r+3r(r-6)=1 equation:


Simplifying
r + 3r(r + -6) = 1

Reorder the terms:
r + 3r(-6 + r) = 1
r + (-6 * 3r + r * 3r) = 1
r + (-18r + 3r2) = 1

Combine like terms: r + -18r = -17r
-17r + 3r2 = 1

Solving
-17r + 3r2 = 1

Solving for variable 'r'.

Reorder the terms:
-1 + -17r + 3r2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -17r + 3r2 = 0

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

Divide each side by '3'.
-0.3333333333 + -5.666666667r + r2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -5.666666667r + 0.3333333333 + r2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -5.666666667r + r2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -5.666666667r + r2 = 0 + 0.3333333333
-5.666666667r + r2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-5.666666667r + r2 = 0.3333333333

The r term is -5.666666667r.  Take half its coefficient (-2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
-5.666666667r + 8.027777782 + r2 = 0.3333333333 + 8.027777782

Reorder the terms:
8.027777782 + -5.666666667r + r2 = 0.3333333333 + 8.027777782

Combine like terms: 0.3333333333 + 8.027777782 = 8.3611111153
8.027777782 + -5.666666667r + r2 = 8.3611111153

Factor a perfect square on the left side:
(r + -2.833333334)(r + -2.833333334) = 8.3611111153

Calculate the square root of the right side: 2.891558596

Break this problem into two subproblems by setting 
(r + -2.833333334) equal to 2.891558596 and -2.891558596.


Subproblem 1
r + -2.833333334 = 2.891558596

Simplifying
r + -2.833333334 = 2.891558596

Reorder the terms:
-2.833333334 + r = 2.891558596

Solving
-2.833333334 + r = 2.891558596

Solving for variable 'r'.

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

Add '2.833333334' to each side of the equation.
-2.833333334 + 2.833333334 + r = 2.891558596 + 2.833333334

Combine like terms: -2.833333334 + 2.833333334 = 0.000000000
0.000000000 + r = 2.891558596 + 2.833333334
r = 2.891558596 + 2.833333334

Combine like terms: 2.891558596 + 2.833333334 = 5.72489193
r = 5.72489193

Simplifying
r = 5.72489193


Subproblem 2
r + -2.833333334 = -2.891558596

Simplifying
r + -2.833333334 = -2.891558596

Reorder the terms:
-2.833333334 + r = -2.891558596

Solving
-2.833333334 + r = -2.891558596

Solving for variable 'r'.

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

Add '2.833333334' to each side of the equation.
-2.833333334 + 2.833333334 + r = -2.891558596 + 2.833333334

Combine like terms: -2.833333334 + 2.833333334 = 0.000000000
0.000000000 + r = -2.891558596 + 2.833333334
r = -2.891558596 + 2.833333334

Combine like terms: -2.891558596 + 2.833333334 = -0.058225262
r = -0.058225262

Simplifying
r = -0.058225262


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {5.72489193, -0.058225262}

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