4x^2+84x+208=0

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Solution for 4x^2+84x+208=0 equation: 


Simplifying
4x2 + 84x + 208 = 0

Reorder the terms:
208 + 84x + 4x2 = 0

Solving
208 + 84x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(52 + 21x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(52 + 21x + x2)' equal to zero and attempt to solve:

Simplifying
52 + 21x + x2 = 0

Solving
52 + 21x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-52' to each side of the equation.
52 + 21x + -52 + x2 = 0 + -52

Reorder the terms:
52 + -52 + 21x + x2 = 0 + -52

Combine like terms: 52 + -52 = 0
0 + 21x + x2 = 0 + -52
21x + x2 = 0 + -52

Combine like terms: 0 + -52 = -52
21x + x2 = -52

The x term is 21x.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21x + 110.25 + x2 = -52 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -52 + 110.25

Combine like terms: -52 + 110.25 = 58.25
110.25 + 21x + x2 = 58.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 58.25

Calculate the square root of the right side: 7.632168761

Break this problem into two subproblems by setting 
(x + 10.5) equal to 7.632168761 and -7.632168761.


Subproblem 1
x + 10.5 = 7.632168761

Simplifying
x + 10.5 = 7.632168761

Reorder the terms:
10.5 + x = 7.632168761

Solving
10.5 + x = 7.632168761

Solving for variable 'x'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = 7.632168761 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 7.632168761 + -10.5
x = 7.632168761 + -10.5

Combine like terms: 7.632168761 + -10.5 = -2.867831239
x = -2.867831239

Simplifying
x = -2.867831239


Subproblem 2
x + 10.5 = -7.632168761

Simplifying
x + 10.5 = -7.632168761

Reorder the terms:
10.5 + x = -7.632168761

Solving
10.5 + x = -7.632168761

Solving for variable 'x'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = -7.632168761 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -7.632168761 + -10.5
x = -7.632168761 + -10.5

Combine like terms: -7.632168761 + -10.5 = -18.132168761
x = -18.132168761

Simplifying
x = -18.132168761


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.867831239, -18.132168761}
Solution
x = {-2.867831239, -18.132168761}

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