x^2+21x+37=0

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Solution for x^2+21x+37=0 equation: 


Simplifying
x2 + 21x + 37 = 0

Reorder the terms:
37 + 21x + x2 = 0

Solving
37 + 21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-37' to each side of the equation.
37 + 21x + -37 + x2 = 0 + -37

Reorder the terms:
37 + -37 + 21x + x2 = 0 + -37

Combine like terms: 37 + -37 = 0
0 + 21x + x2 = 0 + -37
21x + x2 = 0 + -37

Combine like terms: 0 + -37 = -37
21x + x2 = -37

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 = -37 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -37 + 110.25

Combine like terms: -37 + 110.25 = 73.25
110.25 + 21x + x2 = 73.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 73.25

Calculate the square root of the right side: 8.558621384

Break this problem into two subproblems by setting 
(x + 10.5) equal to 8.558621384 and -8.558621384.


Subproblem 1
x + 10.5 = 8.558621384

Simplifying
x + 10.5 = 8.558621384

Reorder the terms:
10.5 + x = 8.558621384

Solving
10.5 + x = 8.558621384

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 = 8.558621384 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 8.558621384 + -10.5
x = 8.558621384 + -10.5

Combine like terms: 8.558621384 + -10.5 = -1.941378616
x = -1.941378616

Simplifying
x = -1.941378616


Subproblem 2
x + 10.5 = -8.558621384

Simplifying
x + 10.5 = -8.558621384

Reorder the terms:
10.5 + x = -8.558621384

Solving
10.5 + x = -8.558621384

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 = -8.558621384 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -8.558621384 + -10.5
x = -8.558621384 + -10.5

Combine like terms: -8.558621384 + -10.5 = -19.058621384
x = -19.058621384

Simplifying
x = -19.058621384


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.941378616, -19.058621384}

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