y^2-4y-23=0

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Solution for y^2-4y-23=0 equation: 


Simplifying
y2 + -4y + -23 = 0

Reorder the terms:
-23 + -4y + y2 = 0

Solving
-23 + -4y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -4y + 23 + y2 = 0 + 23

Reorder the terms:
-23 + 23 + -4y + y2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -4y + y2 = 0 + 23
-4y + y2 = 0 + 23

Combine like terms: 0 + 23 = 23
-4y + y2 = 23

The y term is -4y.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4y + 4 + y2 = 23 + 4

Reorder the terms:
4 + -4y + y2 = 23 + 4

Combine like terms: 23 + 4 = 27
4 + -4y + y2 = 27

Factor a perfect square on the left side:
(y + -2)(y + -2) = 27

Calculate the square root of the right side: 5.196152423

Break this problem into two subproblems by setting 
(y + -2) equal to 5.196152423 and -5.196152423.


Subproblem 1
y + -2 = 5.196152423

Simplifying
y + -2 = 5.196152423

Reorder the terms:
-2 + y = 5.196152423

Solving
-2 + y = 5.196152423

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = 5.196152423 + 2

Combine like terms: -2 + 2 = 0
0 + y = 5.196152423 + 2
y = 5.196152423 + 2

Combine like terms: 5.196152423 + 2 = 7.196152423
y = 7.196152423

Simplifying
y = 7.196152423


Subproblem 2
y + -2 = -5.196152423

Simplifying
y + -2 = -5.196152423

Reorder the terms:
-2 + y = -5.196152423

Solving
-2 + y = -5.196152423

Solving for variable 'y'.

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

Add '2' to each side of the equation.
-2 + 2 + y = -5.196152423 + 2

Combine like terms: -2 + 2 = 0
0 + y = -5.196152423 + 2
y = -5.196152423 + 2

Combine like terms: -5.196152423 + 2 = -3.196152423
y = -3.196152423

Simplifying
y = -3.196152423


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.196152423, -3.196152423}

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