4x^2=6+8x

Solution for 4x^2=6+8x equation:

Simplifying
4x2 = 6 + 8x

Solving
4x2 = 6 + 8x

Solving for variable 'x'.

Reorder the terms:
-6 + -8x + 4x2 = 6 + 8x + -6 + -8x

Reorder the terms:
-6 + -8x + 4x2 = 6 + -6 + 8x + -8x

Combine like terms: 6 + -6 = 0
-6 + -8x + 4x2 = 0 + 8x + -8x
-6 + -8x + 4x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
-6 + -8x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + -4x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + -4x + 2x2)' equal to zero and attempt to solve:

Simplifying
-3 + -4x + 2x2 = 0

Solving
-3 + -4x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + -2x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -2x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -2x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -2x + x2 = 0 + 1.5
-2x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-2x + x2 = 1.5

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 = 1.5 + 1

Reorder the terms:
1 + -2x + x2 = 1.5 + 1

Combine like terms: 1.5 + 1 = 2.5
1 + -2x + x2 = 2.5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(x + -1) equal to 1.58113883 and -1.58113883.

Subproblem 1
x + -1 = 1.58113883

Simplifying
x + -1 = 1.58113883

Reorder the terms:
-1 + x = 1.58113883

Solving
-1 + x = 1.58113883

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 = 1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.58113883 + 1
x = 1.58113883 + 1

Combine like terms: 1.58113883 + 1 = 2.58113883
x = 2.58113883

Simplifying
x = 2.58113883

Subproblem 2
x + -1 = -1.58113883

Simplifying
x + -1 = -1.58113883

Reorder the terms:
-1 + x = -1.58113883

Solving
-1 + x = -1.58113883

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 = -1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.58113883 + 1
x = -1.58113883 + 1

Combine like terms: -1.58113883 + 1 = -0.58113883
x = -0.58113883

Simplifying
x = -0.58113883

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.58113883, -0.58113883}
Solution
x = {2.58113883, -0.58113883}