y^2+y=21

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Solution for y^2+y=21 equation: 


Simplifying
y2 + y = 21

Reorder the terms:
y + y2 = 21

Solving
y + y2 = 21

Solving for variable 'y'.

Reorder the terms:
-21 + y + y2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + y + 21 + y2 = 0 + 21

Reorder the terms:
-21 + 21 + y + y2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + y + y2 = 0 + 21
y + y2 = 0 + 21

Combine like terms: 0 + 21 = 21
y + y2 = 21

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
y + 0.25 + y2 = 21 + 0.25

Reorder the terms:
0.25 + y + y2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + y + y2 = 21.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(y + 0.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
y + 0.5 = 4.609772229

Simplifying
y + 0.5 = 4.609772229

Reorder the terms:
0.5 + y = 4.609772229

Solving
0.5 + y = 4.609772229

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 4.609772229 + -0.5
y = 4.609772229 + -0.5

Combine like terms: 4.609772229 + -0.5 = 4.109772229
y = 4.109772229

Simplifying
y = 4.109772229


Subproblem 2
y + 0.5 = -4.609772229

Simplifying
y + 0.5 = -4.609772229

Reorder the terms:
0.5 + y = -4.609772229

Solving
0.5 + y = -4.609772229

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -4.609772229 + -0.5
y = -4.609772229 + -0.5

Combine like terms: -4.609772229 + -0.5 = -5.109772229
y = -5.109772229

Simplifying
y = -5.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {4.109772229, -5.109772229}

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