6x^2-27x-7=0

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Solution for 6x^2-27x-7=0 equation:


Simplifying
6x2 + -27x + -7 = 0

Reorder the terms:
-7 + -27x + 6x2 = 0

Solving
-7 + -27x + 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'.
-1.166666667 + -4.5x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -4.5x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -4.5x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -4.5x + x2 = 0 + 1.166666667
-4.5x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-4.5x + x2 = 1.166666667

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 1.166666667 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 1.166666667 + 5.0625

Combine like terms: 1.166666667 + 5.0625 = 6.229166667
5.0625 + -4.5x + x2 = 6.229166667

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 6.229166667

Calculate the square root of the right side: 2.495829855

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.495829855 and -2.495829855.

Subproblem 1

x + -2.25 = 2.495829855 Simplifying x + -2.25 = 2.495829855 Reorder the terms: -2.25 + x = 2.495829855 Solving -2.25 + x = 2.495829855 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 2.495829855 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.495829855 + 2.25 x = 2.495829855 + 2.25 Combine like terms: 2.495829855 + 2.25 = 4.745829855 x = 4.745829855 Simplifying x = 4.745829855

Subproblem 2

x + -2.25 = -2.495829855 Simplifying x + -2.25 = -2.495829855 Reorder the terms: -2.25 + x = -2.495829855 Solving -2.25 + x = -2.495829855 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -2.495829855 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.495829855 + 2.25 x = -2.495829855 + 2.25 Combine like terms: -2.495829855 + 2.25 = -0.245829855 x = -0.245829855 Simplifying x = -0.245829855

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.745829855, -0.245829855}

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