x^2-21x+22=0

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Solution for x^2-21x+22=0 equation: 


Simplifying
x2 + -21x + 22 = 0

Reorder the terms:
22 + -21x + x2 = 0

Solving
22 + -21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -21x + -22 + x2 = 0 + -22

Reorder the terms:
22 + -22 + -21x + x2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -21x + x2 = 0 + -22
-21x + x2 = 0 + -22

Combine like terms: 0 + -22 = -22
-21x + x2 = -22

The x term is -21x.  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.
-21x + 110.25 + x2 = -22 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -22 + 110.25

Combine like terms: -22 + 110.25 = 88.25
110.25 + -21x + x2 = 88.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 88.25

Calculate the square root of the right side: 9.394147114

Break this problem into two subproblems by setting 
(x + -10.5) equal to 9.394147114 and -9.394147114.


Subproblem 1
x + -10.5 = 9.394147114

Simplifying
x + -10.5 = 9.394147114

Reorder the terms:
-10.5 + x = 9.394147114

Solving
-10.5 + x = 9.394147114

Solving for variable 'x'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = 9.394147114 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = 9.394147114 + 10.5
x = 9.394147114 + 10.5

Combine like terms: 9.394147114 + 10.5 = 19.894147114
x = 19.894147114

Simplifying
x = 19.894147114


Subproblem 2
x + -10.5 = -9.394147114

Simplifying
x + -10.5 = -9.394147114

Reorder the terms:
-10.5 + x = -9.394147114

Solving
-10.5 + x = -9.394147114

Solving for variable 'x'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = -9.394147114 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = -9.394147114 + 10.5
x = -9.394147114 + 10.5

Combine like terms: -9.394147114 + 10.5 = 1.105852886
x = 1.105852886

Simplifying
x = 1.105852886


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.894147114, 1.105852886}

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