4y^2+22y-5=0

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Solution for 4y^2+22y-5=0 equation: 


Simplifying
4y2 + 22y + -5 = 0

Reorder the terms:
-5 + 22y + 4y2 = 0

Solving
-5 + 22y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
-1.25 + 5.5y + y2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 5.5y + 1.25 + y2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 5.5y + y2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 5.5y + y2 = 0 + 1.25
5.5y + y2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
5.5y + y2 = 1.25

The y term is 5.5y.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5y + 7.5625 + y2 = 1.25 + 7.5625

Reorder the terms:
7.5625 + 5.5y + y2 = 1.25 + 7.5625

Combine like terms: 1.25 + 7.5625 = 8.8125
7.5625 + 5.5y + y2 = 8.8125

Factor a perfect square on the left side:
(y + 2.75)(y + 2.75) = 8.8125

Calculate the square root of the right side: 2.968585522

Break this problem into two subproblems by setting 
(y + 2.75) equal to 2.968585522 and -2.968585522.


Subproblem 1
y + 2.75 = 2.968585522

Simplifying
y + 2.75 = 2.968585522

Reorder the terms:
2.75 + y = 2.968585522

Solving
2.75 + y = 2.968585522

Solving for variable 'y'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + y = 2.968585522 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + y = 2.968585522 + -2.75
y = 2.968585522 + -2.75

Combine like terms: 2.968585522 + -2.75 = 0.218585522
y = 0.218585522

Simplifying
y = 0.218585522


Subproblem 2
y + 2.75 = -2.968585522

Simplifying
y + 2.75 = -2.968585522

Reorder the terms:
2.75 + y = -2.968585522

Solving
2.75 + y = -2.968585522

Solving for variable 'y'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + y = -2.968585522 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + y = -2.968585522 + -2.75
y = -2.968585522 + -2.75

Combine like terms: -2.968585522 + -2.75 = -5.718585522
y = -5.718585522

Simplifying
y = -5.718585522


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.218585522, -5.718585522}

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