x^2-10x+22.5=0

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Solution for x^2-10x+22.5=0 equation: 


Simplifying
x2 + -10x + 22.5 = 0

Reorder the terms:
22.5 + -10x + x2 = 0

Solving
22.5 + -10x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-22.5' to each side of the equation.
22.5 + -10x + -22.5 + x2 = 0 + -22.5

Reorder the terms:
22.5 + -22.5 + -10x + x2 = 0 + -22.5

Combine like terms: 22.5 + -22.5 = 0.0
0.0 + -10x + x2 = 0 + -22.5
-10x + x2 = 0 + -22.5

Combine like terms: 0 + -22.5 = -22.5
-10x + x2 = -22.5

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -22.5 + 25

Reorder the terms:
25 + -10x + x2 = -22.5 + 25

Combine like terms: -22.5 + 25 = 2.5
25 + -10x + x2 = 2.5

Factor a perfect square on the left side:
(x + -5)(x + -5) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(x + -5) equal to 1.58113883 and -1.58113883.


Subproblem 1
x + -5 = 1.58113883

Simplifying
x + -5 = 1.58113883

Reorder the terms:
-5 + x = 1.58113883

Solving
-5 + x = 1.58113883

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = 1.58113883 + 5

Combine like terms: -5 + 5 = 0
0 + x = 1.58113883 + 5
x = 1.58113883 + 5

Combine like terms: 1.58113883 + 5 = 6.58113883
x = 6.58113883

Simplifying
x = 6.58113883


Subproblem 2
x + -5 = -1.58113883

Simplifying
x + -5 = -1.58113883

Reorder the terms:
-5 + x = -1.58113883

Solving
-5 + x = -1.58113883

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = -1.58113883 + 5

Combine like terms: -5 + 5 = 0
0 + x = -1.58113883 + 5
x = -1.58113883 + 5

Combine like terms: -1.58113883 + 5 = 3.41886117
x = 3.41886117

Simplifying
x = 3.41886117


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.58113883, 3.41886117}

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