2(x-5)(x-5)+6=22

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Solution for 2(x-5)(x-5)+6=22 equation: 


Simplifying
2(x + -5)(x + -5) + 6 = 22

Reorder the terms:
2(-5 + x)(x + -5) + 6 = 22

Reorder the terms:
2(-5 + x)(-5 + x) + 6 = 22

Multiply (-5 + x) * (-5 + x)
2(-5(-5 + x) + x(-5 + x)) + 6 = 22
2((-5 * -5 + x * -5) + x(-5 + x)) + 6 = 22
2((25 + -5x) + x(-5 + x)) + 6 = 22
2(25 + -5x + (-5 * x + x * x)) + 6 = 22
2(25 + -5x + (-5x + x2)) + 6 = 22

Combine like terms: -5x + -5x = -10x
2(25 + -10x + x2) + 6 = 22
(25 * 2 + -10x * 2 + x2 * 2) + 6 = 22
(50 + -20x + 2x2) + 6 = 22

Reorder the terms:
50 + 6 + -20x + 2x2 = 22

Combine like terms: 50 + 6 = 56
56 + -20x + 2x2 = 22

Solving
56 + -20x + 2x2 = 22

Solving for variable 'x'.

Reorder the terms:
56 + -22 + -20x + 2x2 = 22 + -22

Combine like terms: 56 + -22 = 34
34 + -20x + 2x2 = 22 + -22

Combine like terms: 22 + -22 = 0
34 + -20x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(17 + -10x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(17 + -10x + x2)' equal to zero and attempt to solve:

Simplifying
17 + -10x + x2 = 0

Solving
17 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-17' to each side of the equation.
17 + -10x + -17 + x2 = 0 + -17

Reorder the terms:
17 + -17 + -10x + x2 = 0 + -17

Combine like terms: 17 + -17 = 0
0 + -10x + x2 = 0 + -17
-10x + x2 = 0 + -17

Combine like terms: 0 + -17 = -17
-10x + x2 = -17

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 = -17 + 25

Reorder the terms:
25 + -10x + x2 = -17 + 25

Combine like terms: -17 + 25 = 8
25 + -10x + x2 = 8

Factor a perfect square on the left side:
(x + -5)(x + -5) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + -5) equal to 2.828427125 and -2.828427125.


Subproblem 1
x + -5 = 2.828427125

Simplifying
x + -5 = 2.828427125

Reorder the terms:
-5 + x = 2.828427125

Solving
-5 + x = 2.828427125

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 = 2.828427125 + 5

Combine like terms: -5 + 5 = 0
0 + x = 2.828427125 + 5
x = 2.828427125 + 5

Combine like terms: 2.828427125 + 5 = 7.828427125
x = 7.828427125

Simplifying
x = 7.828427125


Subproblem 2
x + -5 = -2.828427125

Simplifying
x + -5 = -2.828427125

Reorder the terms:
-5 + x = -2.828427125

Solving
-5 + x = -2.828427125

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 = -2.828427125 + 5

Combine like terms: -5 + 5 = 0
0 + x = -2.828427125 + 5
x = -2.828427125 + 5

Combine like terms: -2.828427125 + 5 = 2.171572875
x = 2.171572875

Simplifying
x = 2.171572875


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.828427125, 2.171572875}
Solution
x = {7.828427125, 2.171572875}

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