y^2-40y+180=0

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Solution for y^2-40y+180=0 equation: 


Simplifying
y2 + -40y + 180 = 0

Reorder the terms:
180 + -40y + y2 = 0

Solving
180 + -40y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-180' to each side of the equation.
180 + -40y + -180 + y2 = 0 + -180

Reorder the terms:
180 + -180 + -40y + y2 = 0 + -180

Combine like terms: 180 + -180 = 0
0 + -40y + y2 = 0 + -180
-40y + y2 = 0 + -180

Combine like terms: 0 + -180 = -180
-40y + y2 = -180

The y term is -40y.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40y + 400 + y2 = -180 + 400

Reorder the terms:
400 + -40y + y2 = -180 + 400

Combine like terms: -180 + 400 = 220
400 + -40y + y2 = 220

Factor a perfect square on the left side:
(y + -20)(y + -20) = 220

Calculate the square root of the right side: 14.832396974

Break this problem into two subproblems by setting 
(y + -20) equal to 14.832396974 and -14.832396974.


Subproblem 1
y + -20 = 14.832396974

Simplifying
y + -20 = 14.832396974

Reorder the terms:
-20 + y = 14.832396974

Solving
-20 + y = 14.832396974

Solving for variable 'y'.

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

Add '20' to each side of the equation.
-20 + 20 + y = 14.832396974 + 20

Combine like terms: -20 + 20 = 0
0 + y = 14.832396974 + 20
y = 14.832396974 + 20

Combine like terms: 14.832396974 + 20 = 34.832396974
y = 34.832396974

Simplifying
y = 34.832396974


Subproblem 2
y + -20 = -14.832396974

Simplifying
y + -20 = -14.832396974

Reorder the terms:
-20 + y = -14.832396974

Solving
-20 + y = -14.832396974

Solving for variable 'y'.

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

Add '20' to each side of the equation.
-20 + 20 + y = -14.832396974 + 20

Combine like terms: -20 + 20 = 0
0 + y = -14.832396974 + 20
y = -14.832396974 + 20

Combine like terms: -14.832396974 + 20 = 5.167603026
y = 5.167603026

Simplifying
y = 5.167603026


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {34.832396974, 5.167603026}

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