Y^2-2Y-42=0

Solution for Y^2-2Y-42=0 equation:

Simplifying
Y2 + -2Y + -42 = 0

Reorder the terms:
-42 + -2Y + Y2 = 0

Solving
-42 + -2Y + Y2 = 0

Solving for variable 'Y'.

Begin completing the square.

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + -2Y + 42 + Y2 = 0 + 42

Reorder the terms:
-42 + 42 + -2Y + Y2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + -2Y + Y2 = 0 + 42
-2Y + Y2 = 0 + 42

Combine like terms: 0 + 42 = 42
-2Y + Y2 = 42

The Y term is -2Y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2Y + 1 + Y2 = 42 + 1

Reorder the terms:
1 + -2Y + Y2 = 42 + 1

Combine like terms: 42 + 1 = 43
1 + -2Y + Y2 = 43

Factor a perfect square on the left side:
(Y + -1)(Y + -1) = 43

Calculate the square root of the right side: 6.557438524

Break this problem into two subproblems by setting 
(Y + -1) equal to 6.557438524 and -6.557438524.

Subproblem 1
Y + -1 = 6.557438524

Simplifying
Y + -1 = 6.557438524

Reorder the terms:
-1 + Y = 6.557438524

Solving
-1 + Y = 6.557438524

Solving for variable 'Y'.

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

Add '1' to each side of the equation.
-1 + 1 + Y = 6.557438524 + 1

Combine like terms: -1 + 1 = 0
0 + Y = 6.557438524 + 1
Y = 6.557438524 + 1

Combine like terms: 6.557438524 + 1 = 7.557438524
Y = 7.557438524

Simplifying
Y = 7.557438524

Subproblem 2
Y + -1 = -6.557438524

Simplifying
Y + -1 = -6.557438524

Reorder the terms:
-1 + Y = -6.557438524

Solving
-1 + Y = -6.557438524

Solving for variable 'Y'.

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

Add '1' to each side of the equation.
-1 + 1 + Y = -6.557438524 + 1

Combine like terms: -1 + 1 = 0
0 + Y = -6.557438524 + 1
Y = -6.557438524 + 1

Combine like terms: -6.557438524 + 1 = -5.557438524
Y = -5.557438524

Simplifying
Y = -5.557438524

Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {7.557438524, -5.557438524}