6x^2-21x-11=0

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Solution for 6x^2-21x-11=0 equation: 


Simplifying
6x2 + -21x + -11 = 0

Reorder the terms:
-11 + -21x + 6x2 = 0

Solving
-11 + -21x + 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'.
-1.833333333 + -3.5x + x2 = 0

Move the constant term to the right:

Add '1.833333333' to each side of the equation.
-1.833333333 + -3.5x + 1.833333333 + x2 = 0 + 1.833333333

Reorder the terms:
-1.833333333 + 1.833333333 + -3.5x + x2 = 0 + 1.833333333

Combine like terms: -1.833333333 + 1.833333333 = 0.000000000
0.000000000 + -3.5x + x2 = 0 + 1.833333333
-3.5x + x2 = 0 + 1.833333333

Combine like terms: 0 + 1.833333333 = 1.833333333
-3.5x + x2 = 1.833333333

The x term is -3.5x.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5x + 3.0625 + x2 = 1.833333333 + 3.0625

Reorder the terms:
3.0625 + -3.5x + x2 = 1.833333333 + 3.0625

Combine like terms: 1.833333333 + 3.0625 = 4.895833333
3.0625 + -3.5x + x2 = 4.895833333

Factor a perfect square on the left side:
(x + -1.75)(x + -1.75) = 4.895833333

Calculate the square root of the right side: 2.212653008

Break this problem into two subproblems by setting 
(x + -1.75) equal to 2.212653008 and -2.212653008.


Subproblem 1
x + -1.75 = 2.212653008

Simplifying
x + -1.75 = 2.212653008

Reorder the terms:
-1.75 + x = 2.212653008

Solving
-1.75 + x = 2.212653008

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = 2.212653008 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = 2.212653008 + 1.75
x = 2.212653008 + 1.75

Combine like terms: 2.212653008 + 1.75 = 3.962653008
x = 3.962653008

Simplifying
x = 3.962653008


Subproblem 2
x + -1.75 = -2.212653008

Simplifying
x + -1.75 = -2.212653008

Reorder the terms:
-1.75 + x = -2.212653008

Solving
-1.75 + x = -2.212653008

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + x = -2.212653008 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + x = -2.212653008 + 1.75
x = -2.212653008 + 1.75

Combine like terms: -2.212653008 + 1.75 = -0.462653008
x = -0.462653008

Simplifying
x = -0.462653008


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.962653008, -0.462653008}

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