4r^2+30r+10=0

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Solution for 4r^2+30r+10=0 equation: 


Simplifying
4r2 + 30r + 10 = 0

Reorder the terms:
10 + 30r + 4r2 = 0

Solving
10 + 30r + 4r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 15r + 2r2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + 15r + 2r2)' equal to zero and attempt to solve:

Simplifying
5 + 15r + 2r2 = 0

Solving
5 + 15r + 2r2 = 0

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

Divide each side by '2'.
2.5 + 7.5r + r2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 7.5r + -2.5 + r2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 7.5r + r2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 7.5r + r2 = 0 + -2.5
7.5r + r2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
7.5r + r2 = -2.5

The r term is 7.5r.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5r + 14.0625 + r2 = -2.5 + 14.0625

Reorder the terms:
14.0625 + 7.5r + r2 = -2.5 + 14.0625

Combine like terms: -2.5 + 14.0625 = 11.5625
14.0625 + 7.5r + r2 = 11.5625

Factor a perfect square on the left side:
(r + 3.75)(r + 3.75) = 11.5625

Calculate the square root of the right side: 3.400367627

Break this problem into two subproblems by setting 
(r + 3.75) equal to 3.400367627 and -3.400367627.


Subproblem 1
r + 3.75 = 3.400367627

Simplifying
r + 3.75 = 3.400367627

Reorder the terms:
3.75 + r = 3.400367627

Solving
3.75 + r = 3.400367627

Solving for variable 'r'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + r = 3.400367627 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + r = 3.400367627 + -3.75
r = 3.400367627 + -3.75

Combine like terms: 3.400367627 + -3.75 = -0.349632373
r = -0.349632373

Simplifying
r = -0.349632373


Subproblem 2
r + 3.75 = -3.400367627

Simplifying
r + 3.75 = -3.400367627

Reorder the terms:
3.75 + r = -3.400367627

Solving
3.75 + r = -3.400367627

Solving for variable 'r'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + r = -3.400367627 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + r = -3.400367627 + -3.75
r = -3.400367627 + -3.75

Combine like terms: -3.400367627 + -3.75 = -7.150367627
r = -7.150367627

Simplifying
r = -7.150367627


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.349632373, -7.150367627}
Solution
r = {-0.349632373, -7.150367627}

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