4x^2-3=x^2-2x+4

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


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

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

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

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

Solving for variable 'x'.

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

Combine like terms: -3 + -4 = -7
-7 + 2x + 4x2 + -1x2 = 4 + -2x + x2 + -4 + 2x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
-7 + 2x + 3x2 = 4 + -2x + x2 + -4 + 2x + -1x2

Reorder the terms:
-7 + 2x + 3x2 = 4 + -4 + -2x + 2x + x2 + -1x2

Combine like terms: 4 + -4 = 0
-7 + 2x + 3x2 = 0 + -2x + 2x + x2 + -1x2
-7 + 2x + 3x2 = -2x + 2x + x2 + -1x2

Combine like terms: -2x + 2x = 0
-7 + 2x + 3x2 = 0 + x2 + -1x2
-7 + 2x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-7 + 2x + 3x2 = 0

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

Divide each side by '3'.
-2.333333333 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + 0.6666666667x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + 0.6666666667x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + 0.6666666667x + x2 = 0 + 2.333333333
0.6666666667x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
0.6666666667x + x2 = 2.333333333

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 2.333333333 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 2.333333333 + 0.1111111112

Combine like terms: 2.333333333 + 0.1111111112 = 2.4444444442
0.1111111112 + 0.6666666667x + x2 = 2.4444444442

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 2.4444444442

Calculate the square root of the right side: 1.56347192

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 1.56347192 and -1.56347192.

Subproblem 1

x + 0.3333333334 = 1.56347192 Simplifying x + 0.3333333334 = 1.56347192 Reorder the terms: 0.3333333334 + x = 1.56347192 Solving 0.3333333334 + x = 1.56347192 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 1.56347192 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 1.56347192 + -0.3333333334 x = 1.56347192 + -0.3333333334 Combine like terms: 1.56347192 + -0.3333333334 = 1.2301385866 x = 1.2301385866 Simplifying x = 1.2301385866

Subproblem 2

x + 0.3333333334 = -1.56347192 Simplifying x + 0.3333333334 = -1.56347192 Reorder the terms: 0.3333333334 + x = -1.56347192 Solving 0.3333333334 + x = -1.56347192 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -1.56347192 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -1.56347192 + -0.3333333334 x = -1.56347192 + -0.3333333334 Combine like terms: -1.56347192 + -0.3333333334 = -1.8968052534 x = -1.8968052534 Simplifying x = -1.8968052534

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.2301385866, -1.8968052534}

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