3a^2-8a-84=0

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Solution for 3a^2-8a-84=0 equation: 


Simplifying
3a2 + -8a + -84 = 0

Reorder the terms:
-84 + -8a + 3a2 = 0

Solving
-84 + -8a + 3a2 = 0

Solving for variable 'a'.

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

Divide each side by '3'.
-28 + -2.666666667a + a2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + -2.666666667a + 28 + a2 = 0 + 28

Reorder the terms:
-28 + 28 + -2.666666667a + a2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + -2.666666667a + a2 = 0 + 28
-2.666666667a + a2 = 0 + 28

Combine like terms: 0 + 28 = 28
-2.666666667a + a2 = 28

The a term is -2.666666667a.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667a + 1.777777780 + a2 = 28 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667a + a2 = 28 + 1.777777780

Combine like terms: 28 + 1.777777780 = 29.77777778
1.777777780 + -2.666666667a + a2 = 29.77777778

Factor a perfect square on the left side:
(a + -1.333333334)(a + -1.333333334) = 29.77777778

Calculate the square root of the right side: 5.456901848

Break this problem into two subproblems by setting 
(a + -1.333333334) equal to 5.456901848 and -5.456901848.


Subproblem 1
a + -1.333333334 = 5.456901848

Simplifying
a + -1.333333334 = 5.456901848

Reorder the terms:
-1.333333334 + a = 5.456901848

Solving
-1.333333334 + a = 5.456901848

Solving for variable 'a'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + a = 5.456901848 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + a = 5.456901848 + 1.333333334
a = 5.456901848 + 1.333333334

Combine like terms: 5.456901848 + 1.333333334 = 6.790235182
a = 6.790235182

Simplifying
a = 6.790235182


Subproblem 2
a + -1.333333334 = -5.456901848

Simplifying
a + -1.333333334 = -5.456901848

Reorder the terms:
-1.333333334 + a = -5.456901848

Solving
-1.333333334 + a = -5.456901848

Solving for variable 'a'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + a = -5.456901848 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + a = -5.456901848 + 1.333333334
a = -5.456901848 + 1.333333334

Combine like terms: -5.456901848 + 1.333333334 = -4.123568514
a = -4.123568514

Simplifying
a = -4.123568514


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {6.790235182, -4.123568514}

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