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 = 3.895833333
5.0625 + 4.5x + x2 = 3.895833333

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 3.895833333

Calculate the square root of the right side: 1.973786547

Break this problem into two subproblems by setting 
(x + 2.25) equal to 1.973786547 and -1.973786547.

Subproblem 1

x + 2.25 = 1.973786547 Simplifying x + 2.25 = 1.973786547 Reorder the terms: 2.25 + x = 1.973786547 Solving 2.25 + x = 1.973786547 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 = 1.973786547 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 1.973786547 + -2.25 x = 1.973786547 + -2.25 Combine like terms: 1.973786547 + -2.25 = -0.276213453 x = -0.276213453 Simplifying x = -0.276213453

Subproblem 2

x + 2.25 = -1.973786547 Simplifying x + 2.25 = -1.973786547 Reorder the terms: 2.25 + x = -1.973786547 Solving 2.25 + x = -1.973786547 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 = -1.973786547 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -1.973786547 + -2.25 x = -1.973786547 + -2.25 Combine like terms: -1.973786547 + -2.25 = -4.223786547 x = -4.223786547 Simplifying x = -4.223786547

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.276213453, -4.223786547}

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