0=2y^2-24y+67

Solution for 0=2y^2-24y+67 equation:

Simplifying
0 = 2y2 + -24y + 67

Reorder the terms:
0 = 67 + -24y + 2y2

Solving
0 = 67 + -24y + 2y2

Solving for variable 'y'.

Combine like terms: 0 + -67 = -67
-67 + 24y + -2y2 = 67 + -24y + 2y2 + -67 + 24y + -2y2

Reorder the terms:
-67 + 24y + -2y2 = 67 + -67 + -24y + 24y + 2y2 + -2y2

Combine like terms: 67 + -67 = 0
-67 + 24y + -2y2 = 0 + -24y + 24y + 2y2 + -2y2
-67 + 24y + -2y2 = -24y + 24y + 2y2 + -2y2

Combine like terms: -24y + 24y = 0
-67 + 24y + -2y2 = 0 + 2y2 + -2y2
-67 + 24y + -2y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
-67 + 24y + -2y2 = 0

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

Divide each side by '-2'.
33.5 + -12y + y2 = 0

Move the constant term to the right:

Add '-33.5' to each side of the equation.
33.5 + -12y + -33.5 + y2 = 0 + -33.5

Reorder the terms:
33.5 + -33.5 + -12y + y2 = 0 + -33.5

Combine like terms: 33.5 + -33.5 = 0.0
0.0 + -12y + y2 = 0 + -33.5
-12y + y2 = 0 + -33.5

Combine like terms: 0 + -33.5 = -33.5
-12y + y2 = -33.5

The y term is -12y.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12y + 36 + y2 = -33.5 + 36

Reorder the terms:
36 + -12y + y2 = -33.5 + 36

Combine like terms: -33.5 + 36 = 2.5
36 + -12y + y2 = 2.5

Factor a perfect square on the left side:
(y + -6)(y + -6) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(y + -6) equal to 1.58113883 and -1.58113883.

Subproblem 1
y + -6 = 1.58113883

Simplifying
y + -6 = 1.58113883

Reorder the terms:
-6 + y = 1.58113883

Solving
-6 + y = 1.58113883

Solving for variable 'y'.

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

Add '6' to each side of the equation.
-6 + 6 + y = 1.58113883 + 6

Combine like terms: -6 + 6 = 0
0 + y = 1.58113883 + 6
y = 1.58113883 + 6

Combine like terms: 1.58113883 + 6 = 7.58113883
y = 7.58113883

Simplifying
y = 7.58113883

Subproblem 2
y + -6 = -1.58113883

Simplifying
y + -6 = -1.58113883

Reorder the terms:
-6 + y = -1.58113883

Solving
-6 + y = -1.58113883

Solving for variable 'y'.

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

Add '6' to each side of the equation.
-6 + 6 + y = -1.58113883 + 6

Combine like terms: -6 + 6 = 0
0 + y = -1.58113883 + 6
y = -1.58113883 + 6

Combine like terms: -1.58113883 + 6 = 4.41886117
y = 4.41886117

Simplifying
y = 4.41886117

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.58113883, 4.41886117}