y^2-9y+13=0

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Solution for y^2-9y+13=0 equation: 


Simplifying
y2 + -9y + 13 = 0

Reorder the terms:
13 + -9y + y2 = 0

Solving
13 + -9y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -9y + -13 + y2 = 0 + -13

Reorder the terms:
13 + -13 + -9y + y2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -9y + y2 = 0 + -13
-9y + y2 = 0 + -13

Combine like terms: 0 + -13 = -13
-9y + y2 = -13

The y term is -9y.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9y + 20.25 + y2 = -13 + 20.25

Reorder the terms:
20.25 + -9y + y2 = -13 + 20.25

Combine like terms: -13 + 20.25 = 7.25
20.25 + -9y + y2 = 7.25

Factor a perfect square on the left side:
(y + -4.5)(y + -4.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + -4.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
y + -4.5 = 2.692582404

Simplifying
y + -4.5 = 2.692582404

Reorder the terms:
-4.5 + y = 2.692582404

Solving
-4.5 + y = 2.692582404

Solving for variable 'y'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + y = 2.692582404 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + y = 2.692582404 + 4.5
y = 2.692582404 + 4.5

Combine like terms: 2.692582404 + 4.5 = 7.192582404
y = 7.192582404

Simplifying
y = 7.192582404


Subproblem 2
y + -4.5 = -2.692582404

Simplifying
y + -4.5 = -2.692582404

Reorder the terms:
-4.5 + y = -2.692582404

Solving
-4.5 + y = -2.692582404

Solving for variable 'y'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + y = -2.692582404 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + y = -2.692582404 + 4.5
y = -2.692582404 + 4.5

Combine like terms: -2.692582404 + 4.5 = 1.807417596
y = 1.807417596

Simplifying
y = 1.807417596


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.192582404, 1.807417596}

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