1.8x^2+2.4x-6.5=0

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Solution for 1.8x^2+2.4x-6.5=0 equation: 


Simplifying
1.8x2 + 2.4x + -6.5 = 0

Reorder the terms:
-6.5 + 2.4x + 1.8x2 = 0

Solving
-6.5 + 2.4x + 1.8x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.8'.
-3.611111111 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '3.611111111' to each side of the equation.
-3.611111111 + 1.333333333x + 3.611111111 + x2 = 0 + 3.611111111

Reorder the terms:
-3.611111111 + 3.611111111 + 1.333333333x + x2 = 0 + 3.611111111

Combine like terms: -3.611111111 + 3.611111111 = 0.000000000
0.000000000 + 1.333333333x + x2 = 0 + 3.611111111
1.333333333x + x2 = 0 + 3.611111111

Combine like terms: 0 + 3.611111111 = 3.611111111
1.333333333x + x2 = 3.611111111

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 3.611111111 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 3.611111111 + 0.4444444442

Combine like terms: 3.611111111 + 0.4444444442 = 4.0555555552
0.4444444442 + 1.333333333x + x2 = 4.0555555552

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 4.0555555552

Calculate the square root of the right side: 2.013840996

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 2.013840996 and -2.013840996.


Subproblem 1
x + 0.6666666665 = 2.013840996

Simplifying
x + 0.6666666665 = 2.013840996

Reorder the terms:
0.6666666665 + x = 2.013840996

Solving
0.6666666665 + x = 2.013840996

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 2.013840996 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 2.013840996 + -0.6666666665
x = 2.013840996 + -0.6666666665

Combine like terms: 2.013840996 + -0.6666666665 = 1.3471743295
x = 1.3471743295

Simplifying
x = 1.3471743295


Subproblem 2
x + 0.6666666665 = -2.013840996

Simplifying
x + 0.6666666665 = -2.013840996

Reorder the terms:
0.6666666665 + x = -2.013840996

Solving
0.6666666665 + x = -2.013840996

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -2.013840996 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -2.013840996 + -0.6666666665
x = -2.013840996 + -0.6666666665

Combine like terms: -2.013840996 + -0.6666666665 = -2.6805076625
x = -2.6805076625

Simplifying
x = -2.6805076625


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.3471743295, -2.6805076625}

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