3r^2-5r=1

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Solution for 3r^2-5r=1 equation: 


Simplifying
3r2 + -5r = 1

Reorder the terms:
-5r + 3r2 = 1

Solving
-5r + 3r2 = 1

Solving for variable 'r'.

Reorder the terms:
-1 + -5r + 3r2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -5r + 3r2 = 0

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

Divide each side by '3'.
-0.3333333333 + -1.666666667r + r2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -1.666666667r + 0.3333333333 + r2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -1.666666667r + r2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -1.666666667r + r2 = 0 + 0.3333333333
-1.666666667r + r2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-1.666666667r + r2 = 0.3333333333

The r term is -1.666666667r.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667r + 0.6944444447 + r2 = 0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667r + r2 = 0.3333333333 + 0.6944444447

Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778
0.6944444447 + -1.666666667r + r2 = 1.027777778

Factor a perfect square on the left side:
(r + -0.8333333335)(r + -0.8333333335) = 1.027777778

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(r + -0.8333333335) equal to 1.013793755 and -1.013793755.


Subproblem 1
r + -0.8333333335 = 1.013793755

Simplifying
r + -0.8333333335 = 1.013793755

Reorder the terms:
-0.8333333335 + r = 1.013793755

Solving
-0.8333333335 + r = 1.013793755

Solving for variable 'r'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + r = 1.013793755 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + r = 1.013793755 + 0.8333333335
r = 1.013793755 + 0.8333333335

Combine like terms: 1.013793755 + 0.8333333335 = 1.8471270885
r = 1.8471270885

Simplifying
r = 1.8471270885


Subproblem 2
r + -0.8333333335 = -1.013793755

Simplifying
r + -0.8333333335 = -1.013793755

Reorder the terms:
-0.8333333335 + r = -1.013793755

Solving
-0.8333333335 + r = -1.013793755

Solving for variable 'r'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + r = -1.013793755 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + r = -1.013793755 + 0.8333333335
r = -1.013793755 + 0.8333333335

Combine like terms: -1.013793755 + 0.8333333335 = -0.1804604215
r = -0.1804604215

Simplifying
r = -0.1804604215


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.8471270885, -0.1804604215}

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