9x^2+3x-4=0

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Solution for 9x^2+3x-4=0 equation: 


Simplifying
9x2 + 3x + -4 = 0

Reorder the terms:
-4 + 3x + 9x2 = 0

Solving
-4 + 3x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-0.4444444444 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 0.3333333333x + 0.4444444444 + x2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 0.3333333333x + x2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.4444444444
0.3333333333x + x2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
0.3333333333x + x2 = 0.4444444444

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 0.4444444444 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.4444444444 + 0.02777777779

Combine like terms: 0.4444444444 + 0.02777777779 = 0.47222222219
0.02777777779 + 0.3333333333x + x2 = 0.47222222219

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 0.47222222219

Calculate the square root of the right side: 0.687184271

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 0.687184271 and -0.687184271.


Subproblem 1
x + 0.1666666667 = 0.687184271

Simplifying
x + 0.1666666667 = 0.687184271

Reorder the terms:
0.1666666667 + x = 0.687184271

Solving
0.1666666667 + x = 0.687184271

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = 0.687184271 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = 0.687184271 + -0.1666666667
x = 0.687184271 + -0.1666666667

Combine like terms: 0.687184271 + -0.1666666667 = 0.5205176043
x = 0.5205176043

Simplifying
x = 0.5205176043


Subproblem 2
x + 0.1666666667 = -0.687184271

Simplifying
x + 0.1666666667 = -0.687184271

Reorder the terms:
0.1666666667 + x = -0.687184271

Solving
0.1666666667 + x = -0.687184271

Solving for variable 'x'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + x = -0.687184271 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + x = -0.687184271 + -0.1666666667
x = -0.687184271 + -0.1666666667

Combine like terms: -0.687184271 + -0.1666666667 = -0.8538509377
x = -0.8538509377

Simplifying
x = -0.8538509377


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.5205176043, -0.8538509377}

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