y^2-23y+46=0

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Solution for y^2-23y+46=0 equation:


Simplifying
y2 + -23y + 46 = 0

Reorder the terms:
46 + -23y + y2 = 0

Solving
46 + -23y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-46' to each side of the equation.
46 + -23y + -46 + y2 = 0 + -46

Reorder the terms:
46 + -46 + -23y + y2 = 0 + -46

Combine like terms: 46 + -46 = 0
0 + -23y + y2 = 0 + -46
-23y + y2 = 0 + -46

Combine like terms: 0 + -46 = -46
-23y + y2 = -46

The y term is -23y.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23y + 132.25 + y2 = -46 + 132.25

Reorder the terms:
132.25 + -23y + y2 = -46 + 132.25

Combine like terms: -46 + 132.25 = 86.25
132.25 + -23y + y2 = 86.25

Factor a perfect square on the left side:
(y + -11.5)(y + -11.5) = 86.25

Calculate the square root of the right side: 9.287087811

Break this problem into two subproblems by setting 
(y + -11.5) equal to 9.287087811 and -9.287087811.


Subproblem 1
y + -11.5 = 9.287087811

Simplifying
y + -11.5 = 9.287087811

Reorder the terms:
-11.5 + y = 9.287087811

Solving
-11.5 + y = 9.287087811

Solving for variable 'y'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + y = 9.287087811 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + y = 9.287087811 + 11.5
y = 9.287087811 + 11.5

Combine like terms: 9.287087811 + 11.5 = 20.787087811
y = 20.787087811

Simplifying
y = 20.787087811


Subproblem 2
y + -11.5 = -9.287087811

Simplifying
y + -11.5 = -9.287087811

Reorder the terms:
-11.5 + y = -9.287087811

Solving
-11.5 + y = -9.287087811

Solving for variable 'y'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + y = -9.287087811 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + y = -9.287087811 + 11.5
y = -9.287087811 + 11.5

Combine like terms: -9.287087811 + 11.5 = 2.212912189
y = 2.212912189

Simplifying
y = 2.212912189


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {20.787087811, 2.212912189}

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