a^2+40+21a=0

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Solution for a^2+40+21a=0 equation: 


Simplifying
a2 + 40 + 21a = 0

Reorder the terms:
40 + 21a + a2 = 0

Solving
40 + 21a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-40' to each side of the equation.
40 + 21a + -40 + a2 = 0 + -40

Reorder the terms:
40 + -40 + 21a + a2 = 0 + -40

Combine like terms: 40 + -40 = 0
0 + 21a + a2 = 0 + -40
21a + a2 = 0 + -40

Combine like terms: 0 + -40 = -40
21a + a2 = -40

The a term is 21a.  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.
21a + 110.25 + a2 = -40 + 110.25

Reorder the terms:
110.25 + 21a + a2 = -40 + 110.25

Combine like terms: -40 + 110.25 = 70.25
110.25 + 21a + a2 = 70.25

Factor a perfect square on the left side:
(a + 10.5)(a + 10.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(a + 10.5) equal to 8.381527307 and -8.381527307.


Subproblem 1
a + 10.5 = 8.381527307

Simplifying
a + 10.5 = 8.381527307

Reorder the terms:
10.5 + a = 8.381527307

Solving
10.5 + a = 8.381527307

Solving for variable 'a'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + a = 8.381527307 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + a = 8.381527307 + -10.5
a = 8.381527307 + -10.5

Combine like terms: 8.381527307 + -10.5 = -2.118472693
a = -2.118472693

Simplifying
a = -2.118472693


Subproblem 2
a + 10.5 = -8.381527307

Simplifying
a + 10.5 = -8.381527307

Reorder the terms:
10.5 + a = -8.381527307

Solving
10.5 + a = -8.381527307

Solving for variable 'a'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + a = -8.381527307 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + a = -8.381527307 + -10.5
a = -8.381527307 + -10.5

Combine like terms: -8.381527307 + -10.5 = -18.881527307
a = -18.881527307

Simplifying
a = -18.881527307


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-2.118472693, -18.881527307}

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