4x=2x^2-5

Solution for 4x=2x^2-5 equation:

Simplifying
4x = 2x2 + -5

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

Solving
4x = -5 + 2x2

Solving for variable 'x'.

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

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

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

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

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

Divide each side by '-2'.
-2.5 + -2x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -2x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -2x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -2x + x2 = 0 + 2.5
-2x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-2x + x2 = 2.5

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 = 2.5 + 1

Reorder the terms:
1 + -2x + x2 = 2.5 + 1

Combine like terms: 2.5 + 1 = 3.5
1 + -2x + x2 = 3.5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3.5

Calculate the square root of the right side: 1.870828693

Break this problem into two subproblems by setting 
(x + -1) equal to 1.870828693 and -1.870828693.

Subproblem 1
x + -1 = 1.870828693

Simplifying
x + -1 = 1.870828693

Reorder the terms:
-1 + x = 1.870828693

Solving
-1 + x = 1.870828693

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 = 1.870828693 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.870828693 + 1
x = 1.870828693 + 1

Combine like terms: 1.870828693 + 1 = 2.870828693
x = 2.870828693

Simplifying
x = 2.870828693

Subproblem 2
x + -1 = -1.870828693

Simplifying
x + -1 = -1.870828693

Reorder the terms:
-1 + x = -1.870828693

Solving
-1 + x = -1.870828693

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 = -1.870828693 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.870828693 + 1
x = -1.870828693 + 1

Combine like terms: -1.870828693 + 1 = -0.870828693
x = -0.870828693

Simplifying
x = -0.870828693

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.870828693, -0.870828693}