r^2+5r+3r=10

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Solution for r^2+5r+3r=10 equation: 


Simplifying
r2 + 5r + 3r = 10

Reorder the terms:
5r + 3r + r2 = 10

Combine like terms: 5r + 3r = 8r
8r + r2 = 10

Solving
8r + r2 = 10

Solving for variable 'r'.

Reorder the terms:
-10 + 8r + r2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 8r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 8r + 10 + r2 = 0 + 10

Reorder the terms:
-10 + 10 + 8r + r2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 8r + r2 = 0 + 10
8r + r2 = 0 + 10

Combine like terms: 0 + 10 = 10
8r + r2 = 10

The r term is 8r.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8r + 16 + r2 = 10 + 16

Reorder the terms:
16 + 8r + r2 = 10 + 16

Combine like terms: 10 + 16 = 26
16 + 8r + r2 = 26

Factor a perfect square on the left side:
(r + 4)(r + 4) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(r + 4) equal to 5.099019514 and -5.099019514.


Subproblem 1
r + 4 = 5.099019514

Simplifying
r + 4 = 5.099019514

Reorder the terms:
4 + r = 5.099019514

Solving
4 + r = 5.099019514

Solving for variable 'r'.

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

Add '-4' to each side of the equation.
4 + -4 + r = 5.099019514 + -4

Combine like terms: 4 + -4 = 0
0 + r = 5.099019514 + -4
r = 5.099019514 + -4

Combine like terms: 5.099019514 + -4 = 1.099019514
r = 1.099019514

Simplifying
r = 1.099019514


Subproblem 2
r + 4 = -5.099019514

Simplifying
r + 4 = -5.099019514

Reorder the terms:
4 + r = -5.099019514

Solving
4 + r = -5.099019514

Solving for variable 'r'.

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

Add '-4' to each side of the equation.
4 + -4 + r = -5.099019514 + -4

Combine like terms: 4 + -4 = 0
0 + r = -5.099019514 + -4
r = -5.099019514 + -4

Combine like terms: -5.099019514 + -4 = -9.099019514
r = -9.099019514

Simplifying
r = -9.099019514


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.099019514, -9.099019514}

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