y^2-21y+18=0

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Solution for y^2-21y+18=0 equation: 


Simplifying
y2 + -21y + 18 = 0

Reorder the terms:
18 + -21y + y2 = 0

Solving
18 + -21y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -21y + -18 + y2 = 0 + -18

Reorder the terms:
18 + -18 + -21y + y2 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -21y + y2 = 0 + -18
-21y + y2 = 0 + -18

Combine like terms: 0 + -18 = -18
-21y + y2 = -18

The y term is -21y.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21y + 110.25 + y2 = -18 + 110.25

Reorder the terms:
110.25 + -21y + y2 = -18 + 110.25

Combine like terms: -18 + 110.25 = 92.25
110.25 + -21y + y2 = 92.25

Factor a perfect square on the left side:
(y + -10.5)(y + -10.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(y + -10.5) equal to 9.604686356 and -9.604686356.


Subproblem 1
y + -10.5 = 9.604686356

Simplifying
y + -10.5 = 9.604686356

Reorder the terms:
-10.5 + y = 9.604686356

Solving
-10.5 + y = 9.604686356

Solving for variable 'y'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + y = 9.604686356 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + y = 9.604686356 + 10.5
y = 9.604686356 + 10.5

Combine like terms: 9.604686356 + 10.5 = 20.104686356
y = 20.104686356

Simplifying
y = 20.104686356


Subproblem 2
y + -10.5 = -9.604686356

Simplifying
y + -10.5 = -9.604686356

Reorder the terms:
-10.5 + y = -9.604686356

Solving
-10.5 + y = -9.604686356

Solving for variable 'y'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + y = -9.604686356 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + y = -9.604686356 + 10.5
y = -9.604686356 + 10.5

Combine like terms: -9.604686356 + 10.5 = 0.895313644
y = 0.895313644

Simplifying
y = 0.895313644


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {20.104686356, 0.895313644}

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