y^2-19y+7=0

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Solution for y^2-19y+7=0 equation: 


Simplifying
y2 + -19y + 7 = 0

Reorder the terms:
7 + -19y + y2 = 0

Solving
7 + -19y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -19y + -7 + y2 = 0 + -7

Reorder the terms:
7 + -7 + -19y + y2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -19y + y2 = 0 + -7
-19y + y2 = 0 + -7

Combine like terms: 0 + -7 = -7
-19y + y2 = -7

The y term is -19y.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19y + 90.25 + y2 = -7 + 90.25

Reorder the terms:
90.25 + -19y + y2 = -7 + 90.25

Combine like terms: -7 + 90.25 = 83.25
90.25 + -19y + y2 = 83.25

Factor a perfect square on the left side:
(y + -9.5)(y + -9.5) = 83.25

Calculate the square root of the right side: 9.124143795

Break this problem into two subproblems by setting 
(y + -9.5) equal to 9.124143795 and -9.124143795.


Subproblem 1
y + -9.5 = 9.124143795

Simplifying
y + -9.5 = 9.124143795

Reorder the terms:
-9.5 + y = 9.124143795

Solving
-9.5 + y = 9.124143795

Solving for variable 'y'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + y = 9.124143795 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + y = 9.124143795 + 9.5
y = 9.124143795 + 9.5

Combine like terms: 9.124143795 + 9.5 = 18.624143795
y = 18.624143795

Simplifying
y = 18.624143795


Subproblem 2
y + -9.5 = -9.124143795

Simplifying
y + -9.5 = -9.124143795

Reorder the terms:
-9.5 + y = -9.124143795

Solving
-9.5 + y = -9.124143795

Solving for variable 'y'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + y = -9.124143795 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + y = -9.124143795 + 9.5
y = -9.124143795 + 9.5

Combine like terms: -9.124143795 + 9.5 = 0.375856205
y = 0.375856205

Simplifying
y = 0.375856205


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {18.624143795, 0.375856205}

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