y^2-6y+7=0

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Solution for y^2-6y+7=0 equation: 


Simplifying
y2 + -6y + 7 = 0

Reorder the terms:
7 + -6y + y2 = 0

Solving
7 + -6y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6y + -7 + y2 = 0 + -7

Reorder the terms:
7 + -7 + -6y + y2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -6y + y2 = 0 + -7
-6y + y2 = 0 + -7

Combine like terms: 0 + -7 = -7
-6y + y2 = -7

The y term is -6y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6y + 9 + y2 = -7 + 9

Reorder the terms:
9 + -6y + y2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + -6y + y2 = 2

Factor a perfect square on the left side:
(y + -3)(y + -3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(y + -3) equal to 1.414213562 and -1.414213562.


Subproblem 1
y + -3 = 1.414213562

Simplifying
y + -3 = 1.414213562

Reorder the terms:
-3 + y = 1.414213562

Solving
-3 + y = 1.414213562

Solving for variable 'y'.

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

Add '3' to each side of the equation.
-3 + 3 + y = 1.414213562 + 3

Combine like terms: -3 + 3 = 0
0 + y = 1.414213562 + 3
y = 1.414213562 + 3

Combine like terms: 1.414213562 + 3 = 4.414213562
y = 4.414213562

Simplifying
y = 4.414213562


Subproblem 2
y + -3 = -1.414213562

Simplifying
y + -3 = -1.414213562

Reorder the terms:
-3 + y = -1.414213562

Solving
-3 + y = -1.414213562

Solving for variable 'y'.

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

Add '3' to each side of the equation.
-3 + 3 + y = -1.414213562 + 3

Combine like terms: -3 + 3 = 0
0 + y = -1.414213562 + 3
y = -1.414213562 + 3

Combine like terms: -1.414213562 + 3 = 1.585786438
y = 1.585786438

Simplifying
y = 1.585786438


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {4.414213562, 1.585786438}

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