2x^2+4x-2.5=0.5x^2+5x+4

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Solution for 2x^2+4x-2.5=0.5x^2+5x+4 equation: 


Simplifying
2x2 + 4x + -2.5 = 0.5x2 + 5x + 4

Reorder the terms:
-2.5 + 4x + 2x2 = 0.5x2 + 5x + 4

Reorder the terms:
-2.5 + 4x + 2x2 = 4 + 5x + 0.5x2

Solving
-2.5 + 4x + 2x2 = 4 + 5x + 0.5x2

Solving for variable 'x'.

Reorder the terms:
-2.5 + -4 + 4x + -5x + 2x2 + -0.5x2 = 4 + 5x + 0.5x2 + -4 + -5x + -0.5x2

Combine like terms: -2.5 + -4 = -6.5
-6.5 + 4x + -5x + 2x2 + -0.5x2 = 4 + 5x + 0.5x2 + -4 + -5x + -0.5x2

Combine like terms: 4x + -5x = -1x
-6.5 + -1x + 2x2 + -0.5x2 = 4 + 5x + 0.5x2 + -4 + -5x + -0.5x2

Combine like terms: 2x2 + -0.5x2 = 1.5x2
-6.5 + -1x + 1.5x2 = 4 + 5x + 0.5x2 + -4 + -5x + -0.5x2

Reorder the terms:
-6.5 + -1x + 1.5x2 = 4 + -4 + 5x + -5x + 0.5x2 + -0.5x2

Combine like terms: 4 + -4 = 0
-6.5 + -1x + 1.5x2 = 0 + 5x + -5x + 0.5x2 + -0.5x2
-6.5 + -1x + 1.5x2 = 5x + -5x + 0.5x2 + -0.5x2

Combine like terms: 5x + -5x = 0
-6.5 + -1x + 1.5x2 = 0 + 0.5x2 + -0.5x2
-6.5 + -1x + 1.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
-6.5 + -1x + 1.5x2 = 0.0

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

Divide each side by '1.5'.
-4.333333333 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '4.333333333' to each side of the equation.
-4.333333333 + -0.6666666667x + 4.333333333 + x2 = 0 + 4.333333333

Reorder the terms:
-4.333333333 + 4.333333333 + -0.6666666667x + x2 = 0 + 4.333333333

Combine like terms: -4.333333333 + 4.333333333 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 4.333333333
-0.6666666667x + x2 = 0 + 4.333333333

Combine like terms: 0 + 4.333333333 = 4.333333333
-0.6666666667x + x2 = 4.333333333

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 = 4.333333333 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 4.333333333 + 0.1111111112

Combine like terms: 4.333333333 + 0.1111111112 = 4.4444444442
0.1111111112 + -0.6666666667x + x2 = 4.4444444442

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 4.4444444442

Calculate the square root of the right side: 2.108185107

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 2.108185107 and -2.108185107.


Subproblem 1
x + -0.3333333334 = 2.108185107

Simplifying
x + -0.3333333334 = 2.108185107

Reorder the terms:
-0.3333333334 + x = 2.108185107

Solving
-0.3333333334 + x = 2.108185107

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 = 2.108185107 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 2.108185107 + 0.3333333334
x = 2.108185107 + 0.3333333334

Combine like terms: 2.108185107 + 0.3333333334 = 2.4415184404
x = 2.4415184404

Simplifying
x = 2.4415184404


Subproblem 2
x + -0.3333333334 = -2.108185107

Simplifying
x + -0.3333333334 = -2.108185107

Reorder the terms:
-0.3333333334 + x = -2.108185107

Solving
-0.3333333334 + x = -2.108185107

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 = -2.108185107 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -2.108185107 + 0.3333333334
x = -2.108185107 + 0.3333333334

Combine like terms: -2.108185107 + 0.3333333334 = -1.7748517736
x = -1.7748517736

Simplifying
x = -1.7748517736


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.4415184404, -1.7748517736}

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