9a^2-13a-21=0

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Solution for 9a^2-13a-21=0 equation: 


Simplifying
9a2 + -13a + -21 = 0

Reorder the terms:
-21 + -13a + 9a2 = 0

Solving
-21 + -13a + 9a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
-2.333333333 + -1.444444444a + a2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -1.444444444a + 2.333333333 + a2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -1.444444444a + a2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -1.444444444a + a2 = 0 + 2.333333333
-1.444444444a + a2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-1.444444444a + a2 = 2.333333333

The a term is -1.444444444a.  Take half its coefficient (-0.722222222).
Square it (0.5216049380) and add it to both sides.

Add '0.5216049380' to each side of the equation.
-1.444444444a + 0.5216049380 + a2 = 2.333333333 + 0.5216049380

Reorder the terms:
0.5216049380 + -1.444444444a + a2 = 2.333333333 + 0.5216049380

Combine like terms: 2.333333333 + 0.5216049380 = 2.854938271
0.5216049380 + -1.444444444a + a2 = 2.854938271

Factor a perfect square on the left side:
(a + -0.722222222)(a + -0.722222222) = 2.854938271

Calculate the square root of the right side: 1.689656258

Break this problem into two subproblems by setting 
(a + -0.722222222) equal to 1.689656258 and -1.689656258.


Subproblem 1
a + -0.722222222 = 1.689656258

Simplifying
a + -0.722222222 = 1.689656258

Reorder the terms:
-0.722222222 + a = 1.689656258

Solving
-0.722222222 + a = 1.689656258

Solving for variable 'a'.

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

Add '0.722222222' to each side of the equation.
-0.722222222 + 0.722222222 + a = 1.689656258 + 0.722222222

Combine like terms: -0.722222222 + 0.722222222 = 0.000000000
0.000000000 + a = 1.689656258 + 0.722222222
a = 1.689656258 + 0.722222222

Combine like terms: 1.689656258 + 0.722222222 = 2.41187848
a = 2.41187848

Simplifying
a = 2.41187848


Subproblem 2
a + -0.722222222 = -1.689656258

Simplifying
a + -0.722222222 = -1.689656258

Reorder the terms:
-0.722222222 + a = -1.689656258

Solving
-0.722222222 + a = -1.689656258

Solving for variable 'a'.

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

Add '0.722222222' to each side of the equation.
-0.722222222 + 0.722222222 + a = -1.689656258 + 0.722222222

Combine like terms: -0.722222222 + 0.722222222 = 0.000000000
0.000000000 + a = -1.689656258 + 0.722222222
a = -1.689656258 + 0.722222222

Combine like terms: -1.689656258 + 0.722222222 = -0.967434036
a = -0.967434036

Simplifying
a = -0.967434036


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.41187848, -0.967434036}

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