10y^2-65y+35=0

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Solution for 10y^2-65y+35=0 equation: 


Simplifying
10y2 + -65y + 35 = 0

Reorder the terms:
35 + -65y + 10y2 = 0

Solving
35 + -65y + 10y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '5'.
5(7 + -13y + 2y2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(7 + -13y + 2y2)' equal to zero and attempt to solve:

Simplifying
7 + -13y + 2y2 = 0

Solving
7 + -13y + 2y2 = 0

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

Divide each side by '2'.
3.5 + -6.5y + y2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -6.5y + -3.5 + y2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -6.5y + y2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -6.5y + y2 = 0 + -3.5
-6.5y + y2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-6.5y + y2 = -3.5

The y term is -6.5y.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5y + 10.5625 + y2 = -3.5 + 10.5625

Reorder the terms:
10.5625 + -6.5y + y2 = -3.5 + 10.5625

Combine like terms: -3.5 + 10.5625 = 7.0625
10.5625 + -6.5y + y2 = 7.0625

Factor a perfect square on the left side:
(y + -3.25)(y + -3.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(y + -3.25) equal to 2.657536453 and -2.657536453.


Subproblem 1
y + -3.25 = 2.657536453

Simplifying
y + -3.25 = 2.657536453

Reorder the terms:
-3.25 + y = 2.657536453

Solving
-3.25 + y = 2.657536453

Solving for variable 'y'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + y = 2.657536453 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + y = 2.657536453 + 3.25
y = 2.657536453 + 3.25

Combine like terms: 2.657536453 + 3.25 = 5.907536453
y = 5.907536453

Simplifying
y = 5.907536453


Subproblem 2
y + -3.25 = -2.657536453

Simplifying
y + -3.25 = -2.657536453

Reorder the terms:
-3.25 + y = -2.657536453

Solving
-3.25 + y = -2.657536453

Solving for variable 'y'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + y = -2.657536453 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + y = -2.657536453 + 3.25
y = -2.657536453 + 3.25

Combine like terms: -2.657536453 + 3.25 = 0.592463547
y = 0.592463547

Simplifying
y = 0.592463547


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.907536453, 0.592463547}
Solution
y = {5.907536453, 0.592463547}

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