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

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

Simplifying
2x2 + -1 = -4x + 1

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

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

Solving
-1 + 2x2 = 1 + -4x

Solving for variable 'x'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 2x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1 + 2x + x2)' equal to zero and attempt to solve:

Simplifying
-1 + 2x + x2 = 0

Solving
-1 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2x + x2 = 0 + 1

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

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

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 1 + 1

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

Combine like terms: 1 + 1 = 2
1 + 2x + x2 = 2

Factor a perfect square on the left side:
(x + 1)(x + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
x + 1 = 1.414213562

Simplifying
x + 1 = 1.414213562

Reorder the terms:
1 + x = 1.414213562

Solving
1 + x = 1.414213562

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + x = 1.414213562 + -1
x = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
x = 0.414213562

Simplifying
x = 0.414213562

Subproblem 2
x + 1 = -1.414213562

Simplifying
x + 1 = -1.414213562

Reorder the terms:
1 + x = -1.414213562

Solving
1 + x = -1.414213562

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + x = -1.414213562 + -1
x = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
x = -2.414213562

Simplifying
x = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.414213562, -2.414213562}
Solution
x = {0.414213562, -2.414213562}