y^2+y-45=0

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Solution for y^2+y-45=0 equation: 


Simplifying
y2 + y + -45 = 0

Reorder the terms:
-45 + y + y2 = 0

Solving
-45 + y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + y + 45 + y2 = 0 + 45

Reorder the terms:
-45 + 45 + y + y2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + y + y2 = 0 + 45
y + y2 = 0 + 45

Combine like terms: 0 + 45 = 45
y + y2 = 45

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 = 45 + 0.25

Reorder the terms:
0.25 + y + y2 = 45 + 0.25

Combine like terms: 45 + 0.25 = 45.25
0.25 + y + y2 = 45.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 45.25

Calculate the square root of the right side: 6.726812024

Break this problem into two subproblems by setting 
(y + 0.5) equal to 6.726812024 and -6.726812024.


Subproblem 1
y + 0.5 = 6.726812024

Simplifying
y + 0.5 = 6.726812024

Reorder the terms:
0.5 + y = 6.726812024

Solving
0.5 + y = 6.726812024

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 = 6.726812024 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 6.726812024 + -0.5
y = 6.726812024 + -0.5

Combine like terms: 6.726812024 + -0.5 = 6.226812024
y = 6.226812024

Simplifying
y = 6.226812024


Subproblem 2
y + 0.5 = -6.726812024

Simplifying
y + 0.5 = -6.726812024

Reorder the terms:
0.5 + y = -6.726812024

Solving
0.5 + y = -6.726812024

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 = -6.726812024 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -6.726812024 + -0.5
y = -6.726812024 + -0.5

Combine like terms: -6.726812024 + -0.5 = -7.226812024
y = -7.226812024

Simplifying
y = -7.226812024


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {6.226812024, -7.226812024}

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