y(y+1)-5=0

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Solution for y(y+1)-5=0 equation: 


Simplifying
y(y + 1) + -5 = 0

Reorder the terms:
y(1 + y) + -5 = 0
(1 * y + y * y) + -5 = 0
(1y + y2) + -5 = 0

Reorder the terms:
-5 + 1y + y2 = 0

Solving
-5 + 1y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 1y + 5 + y2 = 0 + 5

Reorder the terms:
-5 + 5 + 1y + y2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 1y + y2 = 0 + 5
1y + y2 = 0 + 5

Combine like terms: 0 + 5 = 5
1y + y2 = 5

The y term is 1y.  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 = 5 + 0.25

Reorder the terms:
0.25 + y + y2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + y + y2 = 5.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
y + 0.5 = 2.291287847

Simplifying
y + 0.5 = 2.291287847

Reorder the terms:
0.5 + y = 2.291287847

Solving
0.5 + y = 2.291287847

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 = 2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 2.291287847 + -0.5
y = 2.291287847 + -0.5

Combine like terms: 2.291287847 + -0.5 = 1.791287847
y = 1.791287847

Simplifying
y = 1.791287847


Subproblem 2
y + 0.5 = -2.291287847

Simplifying
y + 0.5 = -2.291287847

Reorder the terms:
0.5 + y = -2.291287847

Solving
0.5 + y = -2.291287847

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 = -2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -2.291287847 + -0.5
y = -2.291287847 + -0.5

Combine like terms: -2.291287847 + -0.5 = -2.791287847
y = -2.791287847

Simplifying
y = -2.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.791287847, -2.791287847}

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