9x^2-6x-19=0

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Solution for 9x^2-6x-19=0 equation: 


Simplifying
9x2 + -6x + -19 = 0

Reorder the terms:
-19 + -6x + 9x2 = 0

Solving
-19 + -6x + 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'.
-2.111111111 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '2.111111111' to each side of the equation.
-2.111111111 + -0.6666666667x + 2.111111111 + x2 = 0 + 2.111111111

Reorder the terms:
-2.111111111 + 2.111111111 + -0.6666666667x + x2 = 0 + 2.111111111

Combine like terms: -2.111111111 + 2.111111111 = 0.000000000
0.000000000 + -0.6666666667x + x2 = 0 + 2.111111111
-0.6666666667x + x2 = 0 + 2.111111111

Combine like terms: 0 + 2.111111111 = 2.111111111
-0.6666666667x + x2 = 2.111111111

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 2.111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 2.111111111 + 0.1111111112

Combine like terms: 2.111111111 + 0.1111111112 = 2.2222222222
0.1111111112 + -0.6666666667x + x2 = 2.2222222222

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 2.2222222222

Calculate the square root of the right side: 1.490711985

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.490711985 and -1.490711985.


Subproblem 1
x + -0.3333333334 = 1.490711985

Simplifying
x + -0.3333333334 = 1.490711985

Reorder the terms:
-0.3333333334 + x = 1.490711985

Solving
-0.3333333334 + x = 1.490711985

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 1.490711985 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 1.490711985 + 0.3333333334
x = 1.490711985 + 0.3333333334

Combine like terms: 1.490711985 + 0.3333333334 = 1.8240453184
x = 1.8240453184

Simplifying
x = 1.8240453184


Subproblem 2
x + -0.3333333334 = -1.490711985

Simplifying
x + -0.3333333334 = -1.490711985

Reorder the terms:
-0.3333333334 + x = -1.490711985

Solving
-0.3333333334 + x = -1.490711985

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -1.490711985 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -1.490711985 + 0.3333333334
x = -1.490711985 + 0.3333333334

Combine like terms: -1.490711985 + 0.3333333334 = -1.1573786516
x = -1.1573786516

Simplifying
x = -1.1573786516


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.8240453184, -1.1573786516}

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