x^2+21x+74=0

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Solution for x^2+21x+74=0 equation:


Simplifying
x2 + 21x + 74 = 0

Reorder the terms:
74 + 21x + x2 = 0

Solving
74 + 21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-74' to each side of the equation.
74 + 21x + -74 + x2 = 0 + -74

Reorder the terms:
74 + -74 + 21x + x2 = 0 + -74

Combine like terms: 74 + -74 = 0
0 + 21x + x2 = 0 + -74
21x + x2 = 0 + -74

Combine like terms: 0 + -74 = -74
21x + x2 = -74

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 = -74 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -74 + 110.25

Combine like terms: -74 + 110.25 = 36.25
110.25 + 21x + x2 = 36.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(x + 10.5) equal to 6.020797289 and -6.020797289.


Subproblem 1
x + 10.5 = 6.020797289

Simplifying
x + 10.5 = 6.020797289

Reorder the terms:
10.5 + x = 6.020797289

Solving
10.5 + x = 6.020797289

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 = 6.020797289 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 6.020797289 + -10.5
x = 6.020797289 + -10.5

Combine like terms: 6.020797289 + -10.5 = -4.479202711
x = -4.479202711

Simplifying
x = -4.479202711


Subproblem 2
x + 10.5 = -6.020797289

Simplifying
x + 10.5 = -6.020797289

Reorder the terms:
10.5 + x = -6.020797289

Solving
10.5 + x = -6.020797289

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 = -6.020797289 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -6.020797289 + -10.5
x = -6.020797289 + -10.5

Combine like terms: -6.020797289 + -10.5 = -16.520797289
x = -16.520797289

Simplifying
x = -16.520797289


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.479202711, -16.520797289}

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