4x^2+38x+361=441

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Solution for 4x^2+38x+361=441 equation: 


Simplifying
4x2 + 38x + 361 = 441

Reorder the terms:
361 + 38x + 4x2 = 441

Solving
361 + 38x + 4x2 = 441

Solving for variable 'x'.

Reorder the terms:
361 + -441 + 38x + 4x2 = 441 + -441

Combine like terms: 361 + -441 = -80
-80 + 38x + 4x2 = 441 + -441

Combine like terms: 441 + -441 = 0
-80 + 38x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-40 + 19x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-40 + 19x + 2x2)' equal to zero and attempt to solve:

Simplifying
-40 + 19x + 2x2 = 0

Solving
-40 + 19x + 2x2 = 0

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

Divide each side by '2'.
-20 + 9.5x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 9.5x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 9.5x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 9.5x + x2 = 0 + 20
9.5x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
9.5x + x2 = 20

The x term is 9.5x.  Take half its coefficient (4.75).
Square it (22.5625) and add it to both sides.

Add '22.5625' to each side of the equation.
9.5x + 22.5625 + x2 = 20 + 22.5625

Reorder the terms:
22.5625 + 9.5x + x2 = 20 + 22.5625

Combine like terms: 20 + 22.5625 = 42.5625
22.5625 + 9.5x + x2 = 42.5625

Factor a perfect square on the left side:
(x + 4.75)(x + 4.75) = 42.5625

Calculate the square root of the right side: 6.523994175

Break this problem into two subproblems by setting 
(x + 4.75) equal to 6.523994175 and -6.523994175.


Subproblem 1
x + 4.75 = 6.523994175

Simplifying
x + 4.75 = 6.523994175

Reorder the terms:
4.75 + x = 6.523994175

Solving
4.75 + x = 6.523994175

Solving for variable 'x'.

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

Add '-4.75' to each side of the equation.
4.75 + -4.75 + x = 6.523994175 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + x = 6.523994175 + -4.75
x = 6.523994175 + -4.75

Combine like terms: 6.523994175 + -4.75 = 1.773994175
x = 1.773994175

Simplifying
x = 1.773994175


Subproblem 2
x + 4.75 = -6.523994175

Simplifying
x + 4.75 = -6.523994175

Reorder the terms:
4.75 + x = -6.523994175

Solving
4.75 + x = -6.523994175

Solving for variable 'x'.

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

Add '-4.75' to each side of the equation.
4.75 + -4.75 + x = -6.523994175 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + x = -6.523994175 + -4.75
x = -6.523994175 + -4.75

Combine like terms: -6.523994175 + -4.75 = -11.273994175
x = -11.273994175

Simplifying
x = -11.273994175


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.773994175, -11.273994175}
Solution
x = {1.773994175, -11.273994175}

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