225-7-5r(3r+4)=3(7-4r)

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Solution for 225-7-5r(3r+4)=3(7-4r) equation: 


Simplifying
225 + -7 + -5r(3r + 4) = 3(7 + -4r)

Reorder the terms:
225 + -7 + -5r(4 + 3r) = 3(7 + -4r)
225 + -7 + (4 * -5r + 3r * -5r) = 3(7 + -4r)
225 + -7 + (-20r + -15r2) = 3(7 + -4r)

Combine like terms: 225 + -7 = 218
218 + -20r + -15r2 = 3(7 + -4r)
218 + -20r + -15r2 = (7 * 3 + -4r * 3)
218 + -20r + -15r2 = (21 + -12r)

Solving
218 + -20r + -15r2 = 21 + -12r

Solving for variable 'r'.

Reorder the terms:
218 + -21 + -20r + 12r + -15r2 = 21 + -12r + -21 + 12r

Combine like terms: 218 + -21 = 197
197 + -20r + 12r + -15r2 = 21 + -12r + -21 + 12r

Combine like terms: -20r + 12r = -8r
197 + -8r + -15r2 = 21 + -12r + -21 + 12r

Reorder the terms:
197 + -8r + -15r2 = 21 + -21 + -12r + 12r

Combine like terms: 21 + -21 = 0
197 + -8r + -15r2 = 0 + -12r + 12r
197 + -8r + -15r2 = -12r + 12r

Combine like terms: -12r + 12r = 0
197 + -8r + -15r2 = 0

Begin completing the square.  Divide all terms by
-15 the coefficient of the squared term: 

Divide each side by '-15'.
-13.13333333 + 0.5333333333r + r2 = 0

Move the constant term to the right:

Add '13.13333333' to each side of the equation.
-13.13333333 + 0.5333333333r + 13.13333333 + r2 = 0 + 13.13333333

Reorder the terms:
-13.13333333 + 13.13333333 + 0.5333333333r + r2 = 0 + 13.13333333

Combine like terms: -13.13333333 + 13.13333333 = 0.00000000
0.00000000 + 0.5333333333r + r2 = 0 + 13.13333333
0.5333333333r + r2 = 0 + 13.13333333

Combine like terms: 0 + 13.13333333 = 13.13333333
0.5333333333r + r2 = 13.13333333

The r term is 0.5333333333r.  Take half its coefficient (0.2666666667).
Square it (0.07111111113) and add it to both sides.

Add '0.07111111113' to each side of the equation.
0.5333333333r + 0.07111111113 + r2 = 13.13333333 + 0.07111111113

Reorder the terms:
0.07111111113 + 0.5333333333r + r2 = 13.13333333 + 0.07111111113

Combine like terms: 13.13333333 + 0.07111111113 = 13.20444444113
0.07111111113 + 0.5333333333r + r2 = 13.20444444113

Factor a perfect square on the left side:
(r + 0.2666666667)(r + 0.2666666667) = 13.20444444113

Calculate the square root of the right side: 3.63379202

Break this problem into two subproblems by setting 
(r + 0.2666666667) equal to 3.63379202 and -3.63379202.


Subproblem 1
r + 0.2666666667 = 3.63379202

Simplifying
r + 0.2666666667 = 3.63379202

Reorder the terms:
0.2666666667 + r = 3.63379202

Solving
0.2666666667 + r = 3.63379202

Solving for variable 'r'.

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

Add '-0.2666666667' to each side of the equation.
0.2666666667 + -0.2666666667 + r = 3.63379202 + -0.2666666667

Combine like terms: 0.2666666667 + -0.2666666667 = 0.0000000000
0.0000000000 + r = 3.63379202 + -0.2666666667
r = 3.63379202 + -0.2666666667

Combine like terms: 3.63379202 + -0.2666666667 = 3.3671253533
r = 3.3671253533

Simplifying
r = 3.3671253533


Subproblem 2
r + 0.2666666667 = -3.63379202

Simplifying
r + 0.2666666667 = -3.63379202

Reorder the terms:
0.2666666667 + r = -3.63379202

Solving
0.2666666667 + r = -3.63379202

Solving for variable 'r'.

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

Add '-0.2666666667' to each side of the equation.
0.2666666667 + -0.2666666667 + r = -3.63379202 + -0.2666666667

Combine like terms: 0.2666666667 + -0.2666666667 = 0.0000000000
0.0000000000 + r = -3.63379202 + -0.2666666667
r = -3.63379202 + -0.2666666667

Combine like terms: -3.63379202 + -0.2666666667 = -3.9004586867
r = -3.9004586867

Simplifying
r = -3.9004586867


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.3671253533, -3.9004586867}

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