y^2+5*y=2

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Solution for y^2+5*y=2 equation: 


Simplifying
y2 + 5y = 2

Reorder the terms:
5y + y2 = 2

Solving
5y + y2 = 2

Solving for variable 'y'.

Reorder the terms:
-2 + 5y + y2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 5y + 2 + y2 = 0 + 2

Reorder the terms:
-2 + 2 + 5y + y2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 5y + y2 = 0 + 2
5y + y2 = 0 + 2

Combine like terms: 0 + 2 = 2
5y + y2 = 2

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 = 2 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 2 + 6.25

Combine like terms: 2 + 6.25 = 8.25
6.25 + 5y + y2 = 8.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(y + 2.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
y + 2.5 = 2.872281323

Simplifying
y + 2.5 = 2.872281323

Reorder the terms:
2.5 + y = 2.872281323

Solving
2.5 + y = 2.872281323

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 = 2.872281323 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 2.872281323 + -2.5
y = 2.872281323 + -2.5

Combine like terms: 2.872281323 + -2.5 = 0.372281323
y = 0.372281323

Simplifying
y = 0.372281323


Subproblem 2
y + 2.5 = -2.872281323

Simplifying
y + 2.5 = -2.872281323

Reorder the terms:
2.5 + y = -2.872281323

Solving
2.5 + y = -2.872281323

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 = -2.872281323 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -2.872281323 + -2.5
y = -2.872281323 + -2.5

Combine like terms: -2.872281323 + -2.5 = -5.372281323
y = -5.372281323

Simplifying
y = -5.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.372281323, -5.372281323}

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