4y^2+33y+23=0

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Solution for 4y^2+33y+23=0 equation: 


Simplifying
4y2 + 33y + 23 = 0

Reorder the terms:
23 + 33y + 4y2 = 0

Solving
23 + 33y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
5.75 + 8.25y + y2 = 0

Move the constant term to the right:

Add '-5.75' to each side of the equation.
5.75 + 8.25y + -5.75 + y2 = 0 + -5.75

Reorder the terms:
5.75 + -5.75 + 8.25y + y2 = 0 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + 8.25y + y2 = 0 + -5.75
8.25y + y2 = 0 + -5.75

Combine like terms: 0 + -5.75 = -5.75
8.25y + y2 = -5.75

The y term is 8.25y.  Take half its coefficient (4.125).
Square it (17.015625) and add it to both sides.

Add '17.015625' to each side of the equation.
8.25y + 17.015625 + y2 = -5.75 + 17.015625

Reorder the terms:
17.015625 + 8.25y + y2 = -5.75 + 17.015625

Combine like terms: -5.75 + 17.015625 = 11.265625
17.015625 + 8.25y + y2 = 11.265625

Factor a perfect square on the left side:
(y + 4.125)(y + 4.125) = 11.265625

Calculate the square root of the right side: 3.356430396

Break this problem into two subproblems by setting 
(y + 4.125) equal to 3.356430396 and -3.356430396.


Subproblem 1
y + 4.125 = 3.356430396

Simplifying
y + 4.125 = 3.356430396

Reorder the terms:
4.125 + y = 3.356430396

Solving
4.125 + y = 3.356430396

Solving for variable 'y'.

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

Add '-4.125' to each side of the equation.
4.125 + -4.125 + y = 3.356430396 + -4.125

Combine like terms: 4.125 + -4.125 = 0.000
0.000 + y = 3.356430396 + -4.125
y = 3.356430396 + -4.125

Combine like terms: 3.356430396 + -4.125 = -0.768569604
y = -0.768569604

Simplifying
y = -0.768569604


Subproblem 2
y + 4.125 = -3.356430396

Simplifying
y + 4.125 = -3.356430396

Reorder the terms:
4.125 + y = -3.356430396

Solving
4.125 + y = -3.356430396

Solving for variable 'y'.

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

Add '-4.125' to each side of the equation.
4.125 + -4.125 + y = -3.356430396 + -4.125

Combine like terms: 4.125 + -4.125 = 0.000
0.000 + y = -3.356430396 + -4.125
y = -3.356430396 + -4.125

Combine like terms: -3.356430396 + -4.125 = -7.481430396
y = -7.481430396

Simplifying
y = -7.481430396


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.768569604, -7.481430396}

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