6y^2-24y-23=0

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Solution for 6y^2-24y-23=0 equation: 


Simplifying
6y2 + -24y + -23 = 0

Reorder the terms:
-23 + -24y + 6y2 = 0

Solving
-23 + -24y + 6y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-3.833333333 + -4y + y2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -4y + 3.833333333 + y2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -4y + y2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -4y + y2 = 0 + 3.833333333
-4y + y2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-4y + y2 = 3.833333333

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 = 3.833333333 + 4

Reorder the terms:
4 + -4y + y2 = 3.833333333 + 4

Combine like terms: 3.833333333 + 4 = 7.833333333
4 + -4y + y2 = 7.833333333

Factor a perfect square on the left side:
(y + -2)(y + -2) = 7.833333333

Calculate the square root of the right side: 2.798809271

Break this problem into two subproblems by setting 
(y + -2) equal to 2.798809271 and -2.798809271.


Subproblem 1
y + -2 = 2.798809271

Simplifying
y + -2 = 2.798809271

Reorder the terms:
-2 + y = 2.798809271

Solving
-2 + y = 2.798809271

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 = 2.798809271 + 2

Combine like terms: -2 + 2 = 0
0 + y = 2.798809271 + 2
y = 2.798809271 + 2

Combine like terms: 2.798809271 + 2 = 4.798809271
y = 4.798809271

Simplifying
y = 4.798809271


Subproblem 2
y + -2 = -2.798809271

Simplifying
y + -2 = -2.798809271

Reorder the terms:
-2 + y = -2.798809271

Solving
-2 + y = -2.798809271

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 = -2.798809271 + 2

Combine like terms: -2 + 2 = 0
0 + y = -2.798809271 + 2
y = -2.798809271 + 2

Combine like terms: -2.798809271 + 2 = -0.798809271
y = -0.798809271

Simplifying
y = -0.798809271


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {4.798809271, -0.798809271}

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