y^2+6.7658y-19.0377=0

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Solution for y^2+6.7658y-19.0377=0 equation: 


Simplifying
y2 + 6.7658y + -19.0377 = 0

Reorder the terms:
-19.0377 + 6.7658y + y2 = 0

Solving
-19.0377 + 6.7658y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '19.0377' to each side of the equation.
-19.0377 + 6.7658y + 19.0377 + y2 = 0 + 19.0377

Reorder the terms:
-19.0377 + 19.0377 + 6.7658y + y2 = 0 + 19.0377

Combine like terms: -19.0377 + 19.0377 = 0.0000
0.0000 + 6.7658y + y2 = 0 + 19.0377
6.7658y + y2 = 0 + 19.0377

Combine like terms: 0 + 19.0377 = 19.0377
6.7658y + y2 = 19.0377

The y term is 6.7658y.  Take half its coefficient (3.3829).
Square it (11.44401241) and add it to both sides.

Add '11.44401241' to each side of the equation.
6.7658y + 11.44401241 + y2 = 19.0377 + 11.44401241

Reorder the terms:
11.44401241 + 6.7658y + y2 = 19.0377 + 11.44401241

Combine like terms: 19.0377 + 11.44401241 = 30.48171241
11.44401241 + 6.7658y + y2 = 30.48171241

Factor a perfect square on the left side:
(y + 3.3829)(y + 3.3829) = 30.48171241

Calculate the square root of the right side: 5.52102458

Break this problem into two subproblems by setting 
(y + 3.3829) equal to 5.52102458 and -5.52102458.


Subproblem 1
y + 3.3829 = 5.52102458

Simplifying
y + 3.3829 = 5.52102458

Reorder the terms:
3.3829 + y = 5.52102458

Solving
3.3829 + y = 5.52102458

Solving for variable 'y'.

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

Add '-3.3829' to each side of the equation.
3.3829 + -3.3829 + y = 5.52102458 + -3.3829

Combine like terms: 3.3829 + -3.3829 = 0.0000
0.0000 + y = 5.52102458 + -3.3829
y = 5.52102458 + -3.3829

Combine like terms: 5.52102458 + -3.3829 = 2.13812458
y = 2.13812458

Simplifying
y = 2.13812458


Subproblem 2
y + 3.3829 = -5.52102458

Simplifying
y + 3.3829 = -5.52102458

Reorder the terms:
3.3829 + y = -5.52102458

Solving
3.3829 + y = -5.52102458

Solving for variable 'y'.

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

Add '-3.3829' to each side of the equation.
3.3829 + -3.3829 + y = -5.52102458 + -3.3829

Combine like terms: 3.3829 + -3.3829 = 0.0000
0.0000 + y = -5.52102458 + -3.3829
y = -5.52102458 + -3.3829

Combine like terms: -5.52102458 + -3.3829 = -8.90392458
y = -8.90392458

Simplifying
y = -8.90392458


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.13812458, -8.90392458}

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