2x^2-14x+22.5=0

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Solution for 2x^2-14x+22.5=0 equation: 


Simplifying
2x2 + -14x + 22.5 = 0

Reorder the terms:
22.5 + -14x + 2x2 = 0

Solving
22.5 + -14x + 2x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
11.25 + -7x + x2 = 0

Move the constant term to the right:

Add '-11.25' to each side of the equation.
11.25 + -7x + -11.25 + x2 = 0 + -11.25

Reorder the terms:
11.25 + -11.25 + -7x + x2 = 0 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + -7x + x2 = 0 + -11.25
-7x + x2 = 0 + -11.25

Combine like terms: 0 + -11.25 = -11.25
-7x + x2 = -11.25

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -11.25 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -11.25 + 12.25

Combine like terms: -11.25 + 12.25 = 1
12.25 + -7x + x2 = 1

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 1

Calculate the square root of the right side: 1

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1 and -1.


Subproblem 1
x + -3.5 = 1

Simplifying
x + -3.5 = 1

Reorder the terms:
-3.5 + x = 1

Solving
-3.5 + x = 1

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 1 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 1 + 3.5
x = 1 + 3.5

Combine like terms: 1 + 3.5 = 4.5
x = 4.5

Simplifying
x = 4.5


Subproblem 2
x + -3.5 = -1

Simplifying
x + -3.5 = -1

Reorder the terms:
-3.5 + x = -1

Solving
-3.5 + x = -1

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -1 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -1 + 3.5
x = -1 + 3.5

Combine like terms: -1 + 3.5 = 2.5
x = 2.5

Simplifying
x = 2.5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.5, 2.5}

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