0=x^2+21x+10

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Solution for 0=x^2+21x+10 equation: 


Simplifying
0 = x2 + 21x + 10

Reorder the terms:
0 = 10 + 21x + x2

Solving
0 = 10 + 21x + x2

Solving for variable 'x'.

Combine like terms: 0 + -10 = -10
-10 + -21x + -1x2 = 10 + 21x + x2 + -10 + -21x + -1x2

Reorder the terms:
-10 + -21x + -1x2 = 10 + -10 + 21x + -21x + x2 + -1x2

Combine like terms: 10 + -10 = 0
-10 + -21x + -1x2 = 0 + 21x + -21x + x2 + -1x2
-10 + -21x + -1x2 = 21x + -21x + x2 + -1x2

Combine like terms: 21x + -21x = 0
-10 + -21x + -1x2 = 0 + x2 + -1x2
-10 + -21x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-10 + -21x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(10 + 21x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(10 + 21x + x2)' equal to zero and attempt to solve:

Simplifying
10 + 21x + x2 = 0

Solving
10 + 21x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + 21x + -10 + x2 = 0 + -10

Reorder the terms:
10 + -10 + 21x + x2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 21x + x2 = 0 + -10
21x + x2 = 0 + -10

Combine like terms: 0 + -10 = -10
21x + x2 = -10

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 = -10 + 110.25

Reorder the terms:
110.25 + 21x + x2 = -10 + 110.25

Combine like terms: -10 + 110.25 = 100.25
110.25 + 21x + x2 = 100.25

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 100.25

Calculate the square root of the right side: 10.012492197

Break this problem into two subproblems by setting 
(x + 10.5) equal to 10.012492197 and -10.012492197.


Subproblem 1
x + 10.5 = 10.012492197

Simplifying
x + 10.5 = 10.012492197

Reorder the terms:
10.5 + x = 10.012492197

Solving
10.5 + x = 10.012492197

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 = 10.012492197 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 10.012492197 + -10.5
x = 10.012492197 + -10.5

Combine like terms: 10.012492197 + -10.5 = -0.487507803
x = -0.487507803

Simplifying
x = -0.487507803


Subproblem 2
x + 10.5 = -10.012492197

Simplifying
x + 10.5 = -10.012492197

Reorder the terms:
10.5 + x = -10.012492197

Solving
10.5 + x = -10.012492197

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 = -10.012492197 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -10.012492197 + -10.5
x = -10.012492197 + -10.5

Combine like terms: -10.012492197 + -10.5 = -20.512492197
x = -20.512492197

Simplifying
x = -20.512492197


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.487507803, -20.512492197}
Solution
x = {-0.487507803, -20.512492197}

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