3x^2-27x+19=0

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Solution for 3x^2-27x+19=0 equation:


Simplifying
3x2 + -27x + 19 = 0

Reorder the terms:
19 + -27x + 3x2 = 0

Solving
19 + -27x + 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'.
6.333333333 + -9x + x2 = 0

Move the constant term to the right:

Add '-6.333333333' to each side of the equation.
6.333333333 + -9x + -6.333333333 + x2 = 0 + -6.333333333

Reorder the terms:
6.333333333 + -6.333333333 + -9x + x2 = 0 + -6.333333333

Combine like terms: 6.333333333 + -6.333333333 = 0.000000000
0.000000000 + -9x + x2 = 0 + -6.333333333
-9x + x2 = 0 + -6.333333333

Combine like terms: 0 + -6.333333333 = -6.333333333
-9x + x2 = -6.333333333

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = -6.333333333 + 20.25

Reorder the terms:
20.25 + -9x + x2 = -6.333333333 + 20.25

Combine like terms: -6.333333333 + 20.25 = 13.916666667
20.25 + -9x + x2 = 13.916666667

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 13.916666667

Calculate the square root of the right side: 3.730504881

Break this problem into two subproblems by setting 
(x + -4.5) equal to 3.730504881 and -3.730504881.

Subproblem 1

x + -4.5 = 3.730504881 Simplifying x + -4.5 = 3.730504881 Reorder the terms: -4.5 + x = 3.730504881 Solving -4.5 + x = 3.730504881 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 3.730504881 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 3.730504881 + 4.5 x = 3.730504881 + 4.5 Combine like terms: 3.730504881 + 4.5 = 8.230504881 x = 8.230504881 Simplifying x = 8.230504881

Subproblem 2

x + -4.5 = -3.730504881 Simplifying x + -4.5 = -3.730504881 Reorder the terms: -4.5 + x = -3.730504881 Solving -4.5 + x = -3.730504881 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -3.730504881 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -3.730504881 + 4.5 x = -3.730504881 + 4.5 Combine like terms: -3.730504881 + 4.5 = 0.769495119 x = 0.769495119 Simplifying x = 0.769495119

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.230504881, 0.769495119}

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