y^2-y=10

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Solution for y^2-y=10 equation: 


Simplifying
y2 + -1y = 10

Reorder the terms:
-1y + y2 = 10

Solving
-1y + y2 = 10

Solving for variable 'y'.

Reorder the terms:
-10 + -1y + y2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -1y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -1y + 10 + y2 = 0 + 10

Reorder the terms:
-10 + 10 + -1y + y2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -1y + y2 = 0 + 10
-1y + y2 = 0 + 10

Combine like terms: 0 + 10 = 10
-1y + y2 = 10

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.
-1y + 0.25 + y2 = 10 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 10 + 0.25

Combine like terms: 10 + 0.25 = 10.25
0.25 + -1y + y2 = 10.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(y + -0.5) equal to 3.201562119 and -3.201562119.


Subproblem 1
y + -0.5 = 3.201562119

Simplifying
y + -0.5 = 3.201562119

Reorder the terms:
-0.5 + y = 3.201562119

Solving
-0.5 + y = 3.201562119

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 = 3.201562119 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 3.201562119 + 0.5
y = 3.201562119 + 0.5

Combine like terms: 3.201562119 + 0.5 = 3.701562119
y = 3.701562119

Simplifying
y = 3.701562119


Subproblem 2
y + -0.5 = -3.201562119

Simplifying
y + -0.5 = -3.201562119

Reorder the terms:
-0.5 + y = -3.201562119

Solving
-0.5 + y = -3.201562119

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 = -3.201562119 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -3.201562119 + 0.5
y = -3.201562119 + 0.5

Combine like terms: -3.201562119 + 0.5 = -2.701562119
y = -2.701562119

Simplifying
y = -2.701562119


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.701562119, -2.701562119}

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