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

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


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

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

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

Multiply (-5 + 2x) * (-5 + 2x)
(-5(-5 + 2x) + 2x * (-5 + 2x)) = 2 + (x + -2)(x + -2)
((-5 * -5 + 2x * -5) + 2x * (-5 + 2x)) = 2 + (x + -2)(x + -2)
((25 + -10x) + 2x * (-5 + 2x)) = 2 + (x + -2)(x + -2)
(25 + -10x + (-5 * 2x + 2x * 2x)) = 2 + (x + -2)(x + -2)
(25 + -10x + (-10x + 4x2)) = 2 + (x + -2)(x + -2)

Combine like terms: -10x + -10x = -20x
(25 + -20x + 4x2) = 2 + (x + -2)(x + -2)

Reorder the terms:
25 + -20x + 4x2 = 2 + (-2 + x)(x + -2)

Reorder the terms:
25 + -20x + 4x2 = 2 + (-2 + x)(-2 + x)

Multiply (-2 + x) * (-2 + x)
25 + -20x + 4x2 = 2 + (-2(-2 + x) + x(-2 + x))
25 + -20x + 4x2 = 2 + ((-2 * -2 + x * -2) + x(-2 + x))
25 + -20x + 4x2 = 2 + ((4 + -2x) + x(-2 + x))
25 + -20x + 4x2 = 2 + (4 + -2x + (-2 * x + x * x))
25 + -20x + 4x2 = 2 + (4 + -2x + (-2x + x2))

Combine like terms: -2x + -2x = -4x
25 + -20x + 4x2 = 2 + (4 + -4x + x2)

Combine like terms: 2 + 4 = 6
25 + -20x + 4x2 = 6 + -4x + x2

Solving
25 + -20x + 4x2 = 6 + -4x + x2

Solving for variable 'x'.

Reorder the terms:
25 + -6 + -20x + 4x + 4x2 + -1x2 = 6 + -4x + x2 + -6 + 4x + -1x2

Combine like terms: 25 + -6 = 19
19 + -20x + 4x + 4x2 + -1x2 = 6 + -4x + x2 + -6 + 4x + -1x2

Combine like terms: -20x + 4x = -16x
19 + -16x + 4x2 + -1x2 = 6 + -4x + x2 + -6 + 4x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
19 + -16x + 3x2 = 6 + -4x + x2 + -6 + 4x + -1x2

Reorder the terms:
19 + -16x + 3x2 = 6 + -6 + -4x + 4x + x2 + -1x2

Combine like terms: 6 + -6 = 0
19 + -16x + 3x2 = 0 + -4x + 4x + x2 + -1x2
19 + -16x + 3x2 = -4x + 4x + x2 + -1x2

Combine like terms: -4x + 4x = 0
19 + -16x + 3x2 = 0 + x2 + -1x2
19 + -16x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
19 + -16x + 3x2 = 0

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

Divide each side by '3'.
6.333333333 + -5.333333333x + x2 = 0

Move the constant term to the right:

Add '-6.333333333' to each side of the equation.
6.333333333 + -5.333333333x + -6.333333333 + x2 = 0 + -6.333333333

Reorder the terms:
6.333333333 + -6.333333333 + -5.333333333x + x2 = 0 + -6.333333333

Combine like terms: 6.333333333 + -6.333333333 = 0.000000000
0.000000000 + -5.333333333x + x2 = 0 + -6.333333333
-5.333333333x + x2 = 0 + -6.333333333

Combine like terms: 0 + -6.333333333 = -6.333333333
-5.333333333x + x2 = -6.333333333

The x term is -5.333333333x.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x + 7.111111113 + x2 = -6.333333333 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x + x2 = -6.333333333 + 7.111111113

Combine like terms: -6.333333333 + 7.111111113 = 0.77777778
7.111111113 + -5.333333333x + x2 = 0.77777778

Factor a perfect square on the left side:
(x + -2.666666667)(x + -2.666666667) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(x + -2.666666667) equal to 0.881917105 and -0.881917105.


Subproblem 1
x + -2.666666667 = 0.881917105

Simplifying
x + -2.666666667 = 0.881917105

Reorder the terms:
-2.666666667 + x = 0.881917105

Solving
-2.666666667 + x = 0.881917105

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = 0.881917105 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = 0.881917105 + 2.666666667
x = 0.881917105 + 2.666666667

Combine like terms: 0.881917105 + 2.666666667 = 3.548583772
x = 3.548583772

Simplifying
x = 3.548583772


Subproblem 2
x + -2.666666667 = -0.881917105

Simplifying
x + -2.666666667 = -0.881917105

Reorder the terms:
-2.666666667 + x = -0.881917105

Solving
-2.666666667 + x = -0.881917105

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = -0.881917105 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = -0.881917105 + 2.666666667
x = -0.881917105 + 2.666666667

Combine like terms: -0.881917105 + 2.666666667 = 1.784749562
x = 1.784749562

Simplifying
x = 1.784749562


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.548583772, 1.784749562}

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