Y^2-24Y-252=0

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Solution for Y^2-24Y-252=0 equation: 


Simplifying
Y2 + -24Y + -252 = 0

Reorder the terms:
-252 + -24Y + Y2 = 0

Solving
-252 + -24Y + Y2 = 0

Solving for variable 'Y'.

Begin completing the square.

Move the constant term to the right:

Add '252' to each side of the equation.
-252 + -24Y + 252 + Y2 = 0 + 252

Reorder the terms:
-252 + 252 + -24Y + Y2 = 0 + 252

Combine like terms: -252 + 252 = 0
0 + -24Y + Y2 = 0 + 252
-24Y + Y2 = 0 + 252

Combine like terms: 0 + 252 = 252
-24Y + Y2 = 252

The Y term is -24Y.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24Y + 144 + Y2 = 252 + 144

Reorder the terms:
144 + -24Y + Y2 = 252 + 144

Combine like terms: 252 + 144 = 396
144 + -24Y + Y2 = 396

Factor a perfect square on the left side:
(Y + -12)(Y + -12) = 396

Calculate the square root of the right side: 19.899748742

Break this problem into two subproblems by setting 
(Y + -12) equal to 19.899748742 and -19.899748742.


Subproblem 1
Y + -12 = 19.899748742

Simplifying
Y + -12 = 19.899748742

Reorder the terms:
-12 + Y = 19.899748742

Solving
-12 + Y = 19.899748742

Solving for variable 'Y'.

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

Add '12' to each side of the equation.
-12 + 12 + Y = 19.899748742 + 12

Combine like terms: -12 + 12 = 0
0 + Y = 19.899748742 + 12
Y = 19.899748742 + 12

Combine like terms: 19.899748742 + 12 = 31.899748742
Y = 31.899748742

Simplifying
Y = 31.899748742


Subproblem 2
Y + -12 = -19.899748742

Simplifying
Y + -12 = -19.899748742

Reorder the terms:
-12 + Y = -19.899748742

Solving
-12 + Y = -19.899748742

Solving for variable 'Y'.

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

Add '12' to each side of the equation.
-12 + 12 + Y = -19.899748742 + 12

Combine like terms: -12 + 12 = 0
0 + Y = -19.899748742 + 12
Y = -19.899748742 + 12

Combine like terms: -19.899748742 + 12 = -7.899748742
Y = -7.899748742

Simplifying
Y = -7.899748742


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {31.899748742, -7.899748742}

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