2.25x^2-4x-1=0

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Solution for 2.25x^2-4x-1=0 equation: 


Simplifying
2.25x2 + -4x + -1 = 0

Reorder the terms:
-1 + -4x + 2.25x2 = 0

Solving
-1 + -4x + 2.25x2 = 0

Solving for variable 'x'.

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

Divide each side by '2.25'.
-0.4444444444 + -1.777777778x + x2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + -1.777777778x + 0.4444444444 + x2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + -1.777777778x + x2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + -1.777777778x + x2 = 0 + 0.4444444444
-1.777777778x + x2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
-1.777777778x + x2 = 0.4444444444

The x term is -1.777777778x.  Take half its coefficient (-0.888888889).
Square it (0.7901234570) and add it to both sides.

Add '0.7901234570' to each side of the equation.
-1.777777778x + 0.7901234570 + x2 = 0.4444444444 + 0.7901234570

Reorder the terms:
0.7901234570 + -1.777777778x + x2 = 0.4444444444 + 0.7901234570

Combine like terms: 0.4444444444 + 0.7901234570 = 1.2345679014
0.7901234570 + -1.777777778x + x2 = 1.2345679014

Factor a perfect square on the left side:
(x + -0.888888889)(x + -0.888888889) = 1.2345679014

Calculate the square root of the right side: 1.111111111

Break this problem into two subproblems by setting 
(x + -0.888888889) equal to 1.111111111 and -1.111111111.


Subproblem 1
x + -0.888888889 = 1.111111111

Simplifying
x + -0.888888889 = 1.111111111

Reorder the terms:
-0.888888889 + x = 1.111111111

Solving
-0.888888889 + x = 1.111111111

Solving for variable 'x'.

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

Add '0.888888889' to each side of the equation.
-0.888888889 + 0.888888889 + x = 1.111111111 + 0.888888889

Combine like terms: -0.888888889 + 0.888888889 = 0.000000000
0.000000000 + x = 1.111111111 + 0.888888889
x = 1.111111111 + 0.888888889

Combine like terms: 1.111111111 + 0.888888889 = 2
x = 2

Simplifying
x = 2


Subproblem 2
x + -0.888888889 = -1.111111111

Simplifying
x + -0.888888889 = -1.111111111

Reorder the terms:
-0.888888889 + x = -1.111111111

Solving
-0.888888889 + x = -1.111111111

Solving for variable 'x'.

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

Add '0.888888889' to each side of the equation.
-0.888888889 + 0.888888889 + x = -1.111111111 + 0.888888889

Combine like terms: -0.888888889 + 0.888888889 = 0.000000000
0.000000000 + x = -1.111111111 + 0.888888889
x = -1.111111111 + 0.888888889

Combine like terms: -1.111111111 + 0.888888889 = -0.222222222
x = -0.222222222

Simplifying
x = -0.222222222


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2, -0.222222222}

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