4x^2+22x+11=0

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Solution for 4x^2+22x+11=0 equation: 


Simplifying
4x2 + 22x + 11 = 0

Reorder the terms:
11 + 22x + 4x2 = 0

Solving
11 + 22x + 4x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
2.75 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-2.75' to each side of the equation.
2.75 + 5.5x + -2.75 + x2 = 0 + -2.75

Reorder the terms:
2.75 + -2.75 + 5.5x + x2 = 0 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + 5.5x + x2 = 0 + -2.75
5.5x + x2 = 0 + -2.75

Combine like terms: 0 + -2.75 = -2.75
5.5x + x2 = -2.75

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -2.75 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -2.75 + 7.5625

Combine like terms: -2.75 + 7.5625 = 4.8125
7.5625 + 5.5x + x2 = 4.8125

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 4.8125

Calculate the square root of the right side: 2.193741097

Break this problem into two subproblems by setting 
(x + 2.75) equal to 2.193741097 and -2.193741097.


Subproblem 1
x + 2.75 = 2.193741097

Simplifying
x + 2.75 = 2.193741097

Reorder the terms:
2.75 + x = 2.193741097

Solving
2.75 + x = 2.193741097

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 2.193741097 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 2.193741097 + -2.75
x = 2.193741097 + -2.75

Combine like terms: 2.193741097 + -2.75 = -0.556258903
x = -0.556258903

Simplifying
x = -0.556258903


Subproblem 2
x + 2.75 = -2.193741097

Simplifying
x + 2.75 = -2.193741097

Reorder the terms:
2.75 + x = -2.193741097

Solving
2.75 + x = -2.193741097

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -2.193741097 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -2.193741097 + -2.75
x = -2.193741097 + -2.75

Combine like terms: -2.193741097 + -2.75 = -4.943741097
x = -4.943741097

Simplifying
x = -4.943741097


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.556258903, -4.943741097}

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