2x^2-3=4x-1

Solution for 2x^2-3=4x-1 equation:

Simplifying
2x2 + -3 = 4x + -1

Reorder the terms:
-3 + 2x2 = 4x + -1

Reorder the terms:
-3 + 2x2 = -1 + 4x

Solving
-3 + 2x2 = -1 + 4x

Solving for variable 'x'.

Reorder the terms:
-3 + 1 + -4x + 2x2 = -1 + 4x + 1 + -4x

Combine like terms: -3 + 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 = 2.414213562
x = 2.414213562

Simplifying
x = 2.414213562

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 = -0.414213562
x = -0.414213562

Simplifying
x = -0.414213562

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