x^2=2+x(1-4x)

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Solution for x^2=2+x(1-4x) equation:


Simplifying
x2 = 2 + x(1 + -4x)
x2 = 2 + (1 * x + -4x * x)
x2 = 2 + (1x + -4x2)

Solving
x2 = 2 + 1x + -4x2

Solving for variable 'x'.

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

Combine like terms: x2 + 4x2 = 5x2
-2 + -1x + 5x2 = 2 + 1x + -4x2 + -2 + -1x + 4x2

Reorder the terms:
-2 + -1x + 5x2 = 2 + -2 + 1x + -1x + -4x2 + 4x2

Combine like terms: 2 + -2 = 0
-2 + -1x + 5x2 = 0 + 1x + -1x + -4x2 + 4x2
-2 + -1x + 5x2 = 1x + -1x + -4x2 + 4x2

Combine like terms: 1x + -1x = 0
-2 + -1x + 5x2 = 0 + -4x2 + 4x2
-2 + -1x + 5x2 = -4x2 + 4x2

Combine like terms: -4x2 + 4x2 = 0
-2 + -1x + 5x2 = 0

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

Divide each side by '5'.
-0.4 + -0.2x + x2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + -0.2x + 0.4 + x2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + -0.2x + x2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + -0.2x + x2 = 0 + 0.4
-0.2x + x2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
-0.2x + x2 = 0.4

The x term is -0.2x.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2x + 0.01 + x2 = 0.4 + 0.01

Reorder the terms:
0.01 + -0.2x + x2 = 0.4 + 0.01

Combine like terms: 0.4 + 0.01 = 0.41
0.01 + -0.2x + x2 = 0.41

Factor a perfect square on the left side:
(x + -0.1)(x + -0.1) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(x + -0.1) equal to 0.640312424 and -0.640312424.

Subproblem 1

x + -0.1 = 0.640312424 Simplifying x + -0.1 = 0.640312424 Reorder the terms: -0.1 + x = 0.640312424 Solving -0.1 + x = 0.640312424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 0.640312424 + 0.1 x = 0.640312424 + 0.1 Combine like terms: 0.640312424 + 0.1 = 0.740312424 x = 0.740312424 Simplifying x = 0.740312424

Subproblem 2

x + -0.1 = -0.640312424 Simplifying x + -0.1 = -0.640312424 Reorder the terms: -0.1 + x = -0.640312424 Solving -0.1 + x = -0.640312424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -0.640312424 + 0.1 x = -0.640312424 + 0.1 Combine like terms: -0.640312424 + 0.1 = -0.540312424 x = -0.540312424 Simplifying x = -0.540312424

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.740312424, -0.540312424}

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