y^2+5y-23=37+1

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Solution for y^2+5y-23=37+1 equation: 


Simplifying
y2 + 5y + -23 = 37 + 1

Reorder the terms:
-23 + 5y + y2 = 37 + 1

Combine like terms: 37 + 1 = 38
-23 + 5y + y2 = 38

Solving
-23 + 5y + y2 = 38

Solving for variable 'y'.

Reorder the terms:
-23 + -38 + 5y + y2 = 38 + -38

Combine like terms: -23 + -38 = -61
-61 + 5y + y2 = 38 + -38

Combine like terms: 38 + -38 = 0
-61 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '61' to each side of the equation.
-61 + 5y + 61 + y2 = 0 + 61

Reorder the terms:
-61 + 61 + 5y + y2 = 0 + 61

Combine like terms: -61 + 61 = 0
0 + 5y + y2 = 0 + 61
5y + y2 = 0 + 61

Combine like terms: 0 + 61 = 61
5y + y2 = 61

The y term is 5y.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5y + 6.25 + y2 = 61 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 61 + 6.25

Combine like terms: 61 + 6.25 = 67.25
6.25 + 5y + y2 = 67.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 67.25

Calculate the square root of the right side: 8.200609733

Break this problem into two subproblems by setting 
(y + 2.5) equal to 8.200609733 and -8.200609733.


Subproblem 1
y + 2.5 = 8.200609733

Simplifying
y + 2.5 = 8.200609733

Reorder the terms:
2.5 + y = 8.200609733

Solving
2.5 + y = 8.200609733

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = 8.200609733 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 8.200609733 + -2.5
y = 8.200609733 + -2.5

Combine like terms: 8.200609733 + -2.5 = 5.700609733
y = 5.700609733

Simplifying
y = 5.700609733


Subproblem 2
y + 2.5 = -8.200609733

Simplifying
y + 2.5 = -8.200609733

Reorder the terms:
2.5 + y = -8.200609733

Solving
2.5 + y = -8.200609733

Solving for variable 'y'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + y = -8.200609733 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -8.200609733 + -2.5
y = -8.200609733 + -2.5

Combine like terms: -8.200609733 + -2.5 = -10.700609733
y = -10.700609733

Simplifying
y = -10.700609733


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.700609733, -10.700609733}

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