6x^2+27x+4=0

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


Simplifying
6x2 + 27x + 4 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 4.5x + x2 = 0 + -0.6666666667
4.5x + x2 = 0 + -0.6666666667

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

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 = -0.6666666667 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = -0.6666666667 + 5.0625

Combine like terms: -0.6666666667 + 5.0625 = 4.3958333333
5.0625 + 4.5x + x2 = 4.3958333333

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 4.3958333333

Calculate the square root of the right side: 2.096624271

Break this problem into two subproblems by setting 
(x + 2.25) equal to 2.096624271 and -2.096624271.

Subproblem 1

x + 2.25 = 2.096624271 Simplifying x + 2.25 = 2.096624271 Reorder the terms: 2.25 + x = 2.096624271 Solving 2.25 + x = 2.096624271 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 = 2.096624271 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.096624271 + -2.25 x = 2.096624271 + -2.25 Combine like terms: 2.096624271 + -2.25 = -0.153375729 x = -0.153375729 Simplifying x = -0.153375729

Subproblem 2

x + 2.25 = -2.096624271 Simplifying x + 2.25 = -2.096624271 Reorder the terms: 2.25 + x = -2.096624271 Solving 2.25 + x = -2.096624271 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 = -2.096624271 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.096624271 + -2.25 x = -2.096624271 + -2.25 Combine like terms: -2.096624271 + -2.25 = -4.346624271 x = -4.346624271 Simplifying x = -4.346624271

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.153375729, -4.346624271}

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