x(4x+22)=22

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Solution for x(4x+22)=22 equation: 


Simplifying
x(4x + 22) = 22

Reorder the terms:
x(22 + 4x) = 22
(22 * x + 4x * x) = 22
(22x + 4x2) = 22

Solving
22x + 4x2 = 22

Solving for variable 'x'.

Reorder the terms:
-22 + 22x + 4x2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 22x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + 11x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-11 + 11x + 2x2)' equal to zero and attempt to solve:

Simplifying
-11 + 11x + 2x2 = 0

Solving
-11 + 11x + 2x2 = 0

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

Divide each side by '2'.
-5.5 + 5.5x + x2 = 0

Move the constant term to the right:

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

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

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 5.5x + x2 = 0 + 5.5
5.5x + x2 = 0 + 5.5

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

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 = 5.5 + 7.5625

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

Combine like terms: 5.5 + 7.5625 = 13.0625
7.5625 + 5.5x + x2 = 13.0625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 13.0625

Calculate the square root of the right side: 3.614208074

Break this problem into two subproblems by setting 
(x + 2.75) equal to 3.614208074 and -3.614208074.


Subproblem 1
x + 2.75 = 3.614208074

Simplifying
x + 2.75 = 3.614208074

Reorder the terms:
2.75 + x = 3.614208074

Solving
2.75 + x = 3.614208074

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 = 3.614208074 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 3.614208074 + -2.75
x = 3.614208074 + -2.75

Combine like terms: 3.614208074 + -2.75 = 0.864208074
x = 0.864208074

Simplifying
x = 0.864208074


Subproblem 2
x + 2.75 = -3.614208074

Simplifying
x + 2.75 = -3.614208074

Reorder the terms:
2.75 + x = -3.614208074

Solving
2.75 + x = -3.614208074

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 = -3.614208074 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -3.614208074 + -2.75
x = -3.614208074 + -2.75

Combine like terms: -3.614208074 + -2.75 = -6.364208074
x = -6.364208074

Simplifying
x = -6.364208074


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.864208074, -6.364208074}
Solution
x = {0.864208074, -6.364208074}

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