4r^2=34-10r

Solution for 4r^2=34-10r equation:

Simplifying
4r2 = 34 + -10r

Solving
4r2 = 34 + -10r

Solving for variable 'r'.

Reorder the terms:
-34 + 10r + 4r2 = 34 + -10r + -34 + 10r

Reorder the terms:
-34 + 10r + 4r2 = 34 + -34 + -10r + 10r

Combine like terms: 34 + -34 = 0
-34 + 10r + 4r2 = 0 + -10r + 10r
-34 + 10r + 4r2 = -10r + 10r

Combine like terms: -10r + 10r = 0
-34 + 10r + 4r2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-17 + 5r + 2r2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-17 + 5r + 2r2)' equal to zero and attempt to solve:

Simplifying
-17 + 5r + 2r2 = 0

Solving
-17 + 5r + 2r2 = 0

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

Divide each side by '2'.
-8.5 + 2.5r + r2 = 0

Move the constant term to the right:

Add '8.5' to each side of the equation.
-8.5 + 2.5r + 8.5 + r2 = 0 + 8.5

Reorder the terms:
-8.5 + 8.5 + 2.5r + r2 = 0 + 8.5

Combine like terms: -8.5 + 8.5 = 0.0
0.0 + 2.5r + r2 = 0 + 8.5
2.5r + r2 = 0 + 8.5

Combine like terms: 0 + 8.5 = 8.5
2.5r + r2 = 8.5

The r term is 2.5r.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5r + 1.5625 + r2 = 8.5 + 1.5625

Reorder the terms:
1.5625 + 2.5r + r2 = 8.5 + 1.5625

Combine like terms: 8.5 + 1.5625 = 10.0625
1.5625 + 2.5r + r2 = 10.0625

Factor a perfect square on the left side:
(r + 1.25)(r + 1.25) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(r + 1.25) equal to 3.172144385 and -3.172144385.

Subproblem 1
r + 1.25 = 3.172144385

Simplifying
r + 1.25 = 3.172144385

Reorder the terms:
1.25 + r = 3.172144385

Solving
1.25 + r = 3.172144385

Solving for variable 'r'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + r = 3.172144385 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + r = 3.172144385 + -1.25
r = 3.172144385 + -1.25

Combine like terms: 3.172144385 + -1.25 = 1.922144385
r = 1.922144385

Simplifying
r = 1.922144385

Subproblem 2
r + 1.25 = -3.172144385

Simplifying
r + 1.25 = -3.172144385

Reorder the terms:
1.25 + r = -3.172144385

Solving
1.25 + r = -3.172144385

Solving for variable 'r'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + r = -3.172144385 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + r = -3.172144385 + -1.25
r = -3.172144385 + -1.25

Combine like terms: -3.172144385 + -1.25 = -4.422144385
r = -4.422144385

Simplifying
r = -4.422144385

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.922144385, -4.422144385}
Solution
r = {1.922144385, -4.422144385}