-6x^2+16x-9=0

Solution for -6x^2+16x-9=0 equation:

Simplifying
-6x2 + 16x + -9 = 0

Reorder the terms:
-9 + 16x + -6x2 = 0

Solving
-9 + 16x + -6x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
1.5 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -2.666666667x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -2.666666667x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -2.666666667x + x2 = 0 + -1.5
-2.666666667x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-2.666666667x + x2 = -1.5

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = -1.5 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -1.5 + 1.777777780

Combine like terms: -1.5 + 1.777777780 = 0.27777778
1.777777780 + -2.666666667x + x2 = 0.27777778

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 0.27777778

Calculate the square root of the right side: 0.527046279

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 0.527046279 and -0.527046279.

Subproblem 1
x + -1.333333334 = 0.527046279

Simplifying
x + -1.333333334 = 0.527046279

Reorder the terms:
-1.333333334 + x = 0.527046279

Solving
-1.333333334 + x = 0.527046279

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = 0.527046279 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 0.527046279 + 1.333333334
x = 0.527046279 + 1.333333334

Combine like terms: 0.527046279 + 1.333333334 = 1.860379613
x = 1.860379613

Simplifying
x = 1.860379613

Subproblem 2
x + -1.333333334 = -0.527046279

Simplifying
x + -1.333333334 = -0.527046279

Reorder the terms:
-1.333333334 + x = -0.527046279

Solving
-1.333333334 + x = -0.527046279

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = -0.527046279 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -0.527046279 + 1.333333334
x = -0.527046279 + 1.333333334

Combine like terms: -0.527046279 + 1.333333334 = 0.806287055
x = 0.806287055

Simplifying
x = 0.806287055

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.860379613, 0.806287055}