4r^2+30r=10r-20

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Solution for 4r^2+30r=10r-20 equation: 


Simplifying
4r2 + 30r = 10r + -20

Reorder the terms:
30r + 4r2 = 10r + -20

Reorder the terms:
30r + 4r2 = -20 + 10r

Solving
30r + 4r2 = -20 + 10r

Solving for variable 'r'.

Reorder the terms:
20 + 30r + -10r + 4r2 = -20 + 10r + 20 + -10r

Combine like terms: 30r + -10r = 20r
20 + 20r + 4r2 = -20 + 10r + 20 + -10r

Reorder the terms:
20 + 20r + 4r2 = -20 + 20 + 10r + -10r

Combine like terms: -20 + 20 = 0
20 + 20r + 4r2 = 0 + 10r + -10r
20 + 20r + 4r2 = 10r + -10r

Combine like terms: 10r + -10r = 0
20 + 20r + 4r2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(5 + 5r + r2) = 0

Ignore the factor 4.

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

Simplifying
5 + 5r + r2 = 0

Solving
5 + 5r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 5r + -5 + r2 = 0 + -5

Reorder the terms:
5 + -5 + 5r + r2 = 0 + -5

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

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

The r term is 5r.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5r + 6.25 + r2 = -5 + 6.25

Reorder the terms:
6.25 + 5r + r2 = -5 + 6.25

Combine like terms: -5 + 6.25 = 1.25
6.25 + 5r + r2 = 1.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(r + 2.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
r + 2.5 = 1.118033989

Simplifying
r + 2.5 = 1.118033989

Reorder the terms:
2.5 + r = 1.118033989

Solving
2.5 + r = 1.118033989

Solving for variable 'r'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + r = 1.118033989 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = 1.118033989 + -2.5
r = 1.118033989 + -2.5

Combine like terms: 1.118033989 + -2.5 = -1.381966011
r = -1.381966011

Simplifying
r = -1.381966011


Subproblem 2
r + 2.5 = -1.118033989

Simplifying
r + 2.5 = -1.118033989

Reorder the terms:
2.5 + r = -1.118033989

Solving
2.5 + r = -1.118033989

Solving for variable 'r'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + r = -1.118033989 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = -1.118033989 + -2.5
r = -1.118033989 + -2.5

Combine like terms: -1.118033989 + -2.5 = -3.618033989
r = -3.618033989

Simplifying
r = -3.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-1.381966011, -3.618033989}
Solution
r = {-1.381966011, -3.618033989}

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