7+24y+2y^2=0

Solution for 7+24y+2y^2=0 equation:

Simplifying
7 + 24y + 2y2 = 0

Solving
7 + 24y + 2y2 = 0

Solving for variable 'y'.

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

Divide each side by '2'.
3.5 + 12y + y2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + 12y + -3.5 + y2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + 12y + y2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + 12y + y2 = 0 + -3.5
12y + y2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
12y + y2 = -3.5

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 = -3.5 + 36

Reorder the terms:
36 + 12y + y2 = -3.5 + 36

Combine like terms: -3.5 + 36 = 32.5
36 + 12y + y2 = 32.5

Factor a perfect square on the left side:
(y + 6)(y + 6) = 32.5

Calculate the square root of the right side: 5.700877125

Break this problem into two subproblems by setting 
(y + 6) equal to 5.700877125 and -5.700877125.

Subproblem 1
y + 6 = 5.700877125

Simplifying
y + 6 = 5.700877125

Reorder the terms:
6 + y = 5.700877125

Solving
6 + y = 5.700877125

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 = 5.700877125 + -6

Combine like terms: 6 + -6 = 0
0 + y = 5.700877125 + -6
y = 5.700877125 + -6

Combine like terms: 5.700877125 + -6 = -0.299122875
y = -0.299122875

Simplifying
y = -0.299122875

Subproblem 2
y + 6 = -5.700877125

Simplifying
y + 6 = -5.700877125

Reorder the terms:
6 + y = -5.700877125

Solving
6 + y = -5.700877125

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 = -5.700877125 + -6

Combine like terms: 6 + -6 = 0
0 + y = -5.700877125 + -6
y = -5.700877125 + -6

Combine like terms: -5.700877125 + -6 = -11.700877125
y = -11.700877125

Simplifying
y = -11.700877125

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.299122875, -11.700877125}