0=2+x^2+-4x

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Solution for 0=2+x^2+-4x equation:


Simplifying
0 = 2 + x2 + -4x

Reorder the terms:
0 = 2 + -4x + x2

Solving
0 = 2 + -4x + x2

Solving for variable 'x'.

Combine like terms: 0 + -2 = -2
-2 + 4x + -1x2 = 2 + -4x + x2 + -2 + 4x + -1x2

Reorder the terms:
-2 + 4x + -1x2 = 2 + -2 + -4x + 4x + x2 + -1x2

Combine like terms: 2 + -2 = 0
-2 + 4x + -1x2 = 0 + -4x + 4x + x2 + -1x2
-2 + 4x + -1x2 = -4x + 4x + x2 + -1x2

Combine like terms: -4x + 4x = 0
-2 + 4x + -1x2 = 0 + x2 + -1x2
-2 + 4x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-2 + 4x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
2 + -4x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + -4x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4x + x2 = 0 + -2
-4x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
-4x + x2 = -2

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = -2 + 4

Reorder the terms:
4 + -4x + x2 = -2 + 4

Combine like terms: -2 + 4 = 2
4 + -4x + x2 = 2

Factor a perfect square on the left side:
(x + -2)(x + -2) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -2) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -2 = 1.414213562 Simplifying x + -2 = 1.414213562 Reorder the terms: -2 + x = 1.414213562 Solving -2 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + x = 1.414213562 + 2 x = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 x = 3.414213562 Simplifying x = 3.414213562

Subproblem 2

x + -2 = -1.414213562 Simplifying x + -2 = -1.414213562 Reorder the terms: -2 + x = -1.414213562 Solving -2 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + x = -1.414213562 + 2 x = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 x = 0.585786438 Simplifying x = 0.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.414213562, 0.585786438}

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