4x^2+42x+22=0

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Solution for 4x^2+42x+22=0 equation: 


Simplifying
4x2 + 42x + 22 = 0

Reorder the terms:
22 + 42x + 4x2 = 0

Solving
22 + 42x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(11 + 21x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(11 + 21x + 2x2)' equal to zero and attempt to solve:

Simplifying
11 + 21x + 2x2 = 0

Solving
11 + 21x + 2x2 = 0

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

Divide each side by '2'.
5.5 + 10.5x + x2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + 10.5x + -5.5 + x2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + 10.5x + x2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + 10.5x + x2 = 0 + -5.5
10.5x + x2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
10.5x + x2 = -5.5

The x term is 10.5x.  Take half its coefficient (5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
10.5x + 27.5625 + x2 = -5.5 + 27.5625

Reorder the terms:
27.5625 + 10.5x + x2 = -5.5 + 27.5625

Combine like terms: -5.5 + 27.5625 = 22.0625
27.5625 + 10.5x + x2 = 22.0625

Factor a perfect square on the left side:
(x + 5.25)(x + 5.25) = 22.0625

Calculate the square root of the right side: 4.697073557

Break this problem into two subproblems by setting 
(x + 5.25) equal to 4.697073557 and -4.697073557.


Subproblem 1
x + 5.25 = 4.697073557

Simplifying
x + 5.25 = 4.697073557

Reorder the terms:
5.25 + x = 4.697073557

Solving
5.25 + x = 4.697073557

Solving for variable 'x'.

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

Add '-5.25' to each side of the equation.
5.25 + -5.25 + x = 4.697073557 + -5.25

Combine like terms: 5.25 + -5.25 = 0.00
0.00 + x = 4.697073557 + -5.25
x = 4.697073557 + -5.25

Combine like terms: 4.697073557 + -5.25 = -0.552926443
x = -0.552926443

Simplifying
x = -0.552926443


Subproblem 2
x + 5.25 = -4.697073557

Simplifying
x + 5.25 = -4.697073557

Reorder the terms:
5.25 + x = -4.697073557

Solving
5.25 + x = -4.697073557

Solving for variable 'x'.

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

Add '-5.25' to each side of the equation.
5.25 + -5.25 + x = -4.697073557 + -5.25

Combine like terms: 5.25 + -5.25 = 0.00
0.00 + x = -4.697073557 + -5.25
x = -4.697073557 + -5.25

Combine like terms: -4.697073557 + -5.25 = -9.947073557
x = -9.947073557

Simplifying
x = -9.947073557


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.552926443, -9.947073557}
Solution
x = {-0.552926443, -9.947073557}

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