4r^2-18r+10=0

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Solution for 4r^2-18r+10=0 equation: 


Simplifying
4r2 + -18r + 10 = 0

Reorder the terms:
10 + -18r + 4r2 = 0

Solving
10 + -18r + 4r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + -9r + 2r2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + -9r + 2r2)' equal to zero and attempt to solve:

Simplifying
5 + -9r + 2r2 = 0

Solving
5 + -9r + 2r2 = 0

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

Divide each side by '2'.
2.5 + -4.5r + r2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + -4.5r + -2.5 + r2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + -4.5r + r2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + -4.5r + r2 = 0 + -2.5
-4.5r + r2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
-4.5r + r2 = -2.5

The r term is -4.5r.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5r + 5.0625 + r2 = -2.5 + 5.0625

Reorder the terms:
5.0625 + -4.5r + r2 = -2.5 + 5.0625

Combine like terms: -2.5 + 5.0625 = 2.5625
5.0625 + -4.5r + r2 = 2.5625

Factor a perfect square on the left side:
(r + -2.25)(r + -2.25) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(r + -2.25) equal to 1.600781059 and -1.600781059.


Subproblem 1
r + -2.25 = 1.600781059

Simplifying
r + -2.25 = 1.600781059

Reorder the terms:
-2.25 + r = 1.600781059

Solving
-2.25 + r = 1.600781059

Solving for variable 'r'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + r = 1.600781059 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + r = 1.600781059 + 2.25
r = 1.600781059 + 2.25

Combine like terms: 1.600781059 + 2.25 = 3.850781059
r = 3.850781059

Simplifying
r = 3.850781059


Subproblem 2
r + -2.25 = -1.600781059

Simplifying
r + -2.25 = -1.600781059

Reorder the terms:
-2.25 + r = -1.600781059

Solving
-2.25 + r = -1.600781059

Solving for variable 'r'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + r = -1.600781059 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + r = -1.600781059 + 2.25
r = -1.600781059 + 2.25

Combine like terms: -1.600781059 + 2.25 = 0.649218941
r = 0.649218941

Simplifying
r = 0.649218941


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.850781059, 0.649218941}
Solution
r = {3.850781059, 0.649218941}

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