6x^2-4x-27=0

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Solution for 6x^2-4x-27=0 equation:


Simplifying
6x2 + -4x + -27 = 0

Reorder the terms:
-27 + -4x + 6x2 = 0

Solving
-27 + -4x + 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'.
-4.5 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -0.6666666667x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -0.6666666667x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -0.6666666667x + x2 = 0 + 4.5
-0.6666666667x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-0.6666666667x + x2 = 4.5

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 = 4.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 4.5 + 0.1111111112

Combine like terms: 4.5 + 0.1111111112 = 4.6111111112
0.1111111112 + -0.6666666667x + x2 = 4.6111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 4.6111111112

Calculate the square root of the right side: 2.147349788

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 2.147349788 and -2.147349788.

Subproblem 1

x + -0.3333333334 = 2.147349788 Simplifying x + -0.3333333334 = 2.147349788 Reorder the terms: -0.3333333334 + x = 2.147349788 Solving -0.3333333334 + x = 2.147349788 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 = 2.147349788 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 2.147349788 + 0.3333333334 x = 2.147349788 + 0.3333333334 Combine like terms: 2.147349788 + 0.3333333334 = 2.4806831214 x = 2.4806831214 Simplifying x = 2.4806831214

Subproblem 2

x + -0.3333333334 = -2.147349788 Simplifying x + -0.3333333334 = -2.147349788 Reorder the terms: -0.3333333334 + x = -2.147349788 Solving -0.3333333334 + x = -2.147349788 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 = -2.147349788 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -2.147349788 + 0.3333333334 x = -2.147349788 + 0.3333333334 Combine like terms: -2.147349788 + 0.3333333334 = -1.8140164546 x = -1.8140164546 Simplifying x = -1.8140164546

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.4806831214, -1.8140164546}

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