-3x^2+16x-9=0

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Solution for -3x^2+16x-9=0 equation: 


Simplifying
-3x2 + 16x + -9 = 0

Reorder the terms:
-9 + 16x + -3x2 = 0

Solving
-9 + 16x + -3x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
3 + -5.333333333x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -5.333333333x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + -5.333333333x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -5.333333333x + x2 = 0 + -3
-5.333333333x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
-5.333333333x + x2 = -3

The x term is -5.333333333x.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x + 7.111111113 + x2 = -3 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x + x2 = -3 + 7.111111113

Combine like terms: -3 + 7.111111113 = 4.111111113
7.111111113 + -5.333333333x + x2 = 4.111111113

Factor a perfect square on the left side:
(x + -2.666666667)(x + -2.666666667) = 4.111111113

Calculate the square root of the right side: 2.027587511

Break this problem into two subproblems by setting 
(x + -2.666666667) equal to 2.027587511 and -2.027587511.


Subproblem 1
x + -2.666666667 = 2.027587511

Simplifying
x + -2.666666667 = 2.027587511

Reorder the terms:
-2.666666667 + x = 2.027587511

Solving
-2.666666667 + x = 2.027587511

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = 2.027587511 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = 2.027587511 + 2.666666667
x = 2.027587511 + 2.666666667

Combine like terms: 2.027587511 + 2.666666667 = 4.694254178
x = 4.694254178

Simplifying
x = 4.694254178


Subproblem 2
x + -2.666666667 = -2.027587511

Simplifying
x + -2.666666667 = -2.027587511

Reorder the terms:
-2.666666667 + x = -2.027587511

Solving
-2.666666667 + x = -2.027587511

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = -2.027587511 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = -2.027587511 + 2.666666667
x = -2.027587511 + 2.666666667

Combine like terms: -2.027587511 + 2.666666667 = 0.639079156
x = 0.639079156

Simplifying
x = 0.639079156


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.694254178, 0.639079156}

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