324=84(12-a)(a-7)

Solution for 324=84(12-a)(a-7) equation:

Simplifying
324 = 84(12 + -1a)(a + -7)

Reorder the terms:
324 = 84(12 + -1a)(-7 + a)

Multiply (12 + -1a) * (-7 + a)
324 = 84(12(-7 + a) + -1a * (-7 + a))
324 = 84((-7 * 12 + a * 12) + -1a * (-7 + a))
324 = 84((-84 + 12a) + -1a * (-7 + a))
324 = 84(-84 + 12a + (-7 * -1a + a * -1a))
324 = 84(-84 + 12a + (7a + -1a2))

Combine like terms: 12a + 7a = 19a
324 = 84(-84 + 19a + -1a2)
324 = (-84 * 84 + 19a * 84 + -1a2 * 84)
324 = (-7056 + 1596a + -84a2)

Solving
324 = -7056 + 1596a + -84a2

Solving for variable 'a'.

Combine like terms: 324 + 7056 = 7380
7380 + -1596a + 84a2 = -7056 + 1596a + -84a2 + 7056 + -1596a + 84a2

Reorder the terms:
7380 + -1596a + 84a2 = -7056 + 7056 + 1596a + -1596a + -84a2 + 84a2

Combine like terms: -7056 + 7056 = 0
7380 + -1596a + 84a2 = 0 + 1596a + -1596a + -84a2 + 84a2
7380 + -1596a + 84a2 = 1596a + -1596a + -84a2 + 84a2

Combine like terms: 1596a + -1596a = 0
7380 + -1596a + 84a2 = 0 + -84a2 + 84a2
7380 + -1596a + 84a2 = -84a2 + 84a2

Combine like terms: -84a2 + 84a2 = 0
7380 + -1596a + 84a2 = 0

Factor out the Greatest Common Factor (GCF), '12'.
12(615 + -133a + 7a2) = 0

Ignore the factor 12.
Subproblem 1
Set the factor '(615 + -133a + 7a2)' equal to zero and attempt to solve:

Simplifying
615 + -133a + 7a2 = 0

Solving
615 + -133a + 7a2 = 0

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

Divide each side by '7'.
87.85714286 + -19a + a2 = 0

Move the constant term to the right:

Add '-87.85714286' to each side of the equation.
87.85714286 + -19a + -87.85714286 + a2 = 0 + -87.85714286

Reorder the terms:
87.85714286 + -87.85714286 + -19a + a2 = 0 + -87.85714286

Combine like terms: 87.85714286 + -87.85714286 = 0.00000000
0.00000000 + -19a + a2 = 0 + -87.85714286
-19a + a2 = 0 + -87.85714286

Combine like terms: 0 + -87.85714286 = -87.85714286
-19a + a2 = -87.85714286

The a term is -19a.  Take half its coefficient (-9.5).
Square it (90.25) and add it to both sides.

Add '90.25' to each side of the equation.
-19a + 90.25 + a2 = -87.85714286 + 90.25

Reorder the terms:
90.25 + -19a + a2 = -87.85714286 + 90.25

Combine like terms: -87.85714286 + 90.25 = 2.39285714
90.25 + -19a + a2 = 2.39285714

Factor a perfect square on the left side:
(a + -9.5)(a + -9.5) = 2.39285714

Calculate the square root of the right side: 1.546886272

Break this problem into two subproblems by setting 
(a + -9.5) equal to 1.546886272 and -1.546886272.

Subproblem 1
a + -9.5 = 1.546886272

Simplifying
a + -9.5 = 1.546886272

Reorder the terms:
-9.5 + a = 1.546886272

Solving
-9.5 + a = 1.546886272

Solving for variable 'a'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + a = 1.546886272 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + a = 1.546886272 + 9.5
a = 1.546886272 + 9.5

Combine like terms: 1.546886272 + 9.5 = 11.046886272
a = 11.046886272

Simplifying
a = 11.046886272

Subproblem 2
a + -9.5 = -1.546886272

Simplifying
a + -9.5 = -1.546886272

Reorder the terms:
-9.5 + a = -1.546886272

Solving
-9.5 + a = -1.546886272

Solving for variable 'a'.

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

Add '9.5' to each side of the equation.
-9.5 + 9.5 + a = -1.546886272 + 9.5

Combine like terms: -9.5 + 9.5 = 0.0
0.0 + a = -1.546886272 + 9.5
a = -1.546886272 + 9.5

Combine like terms: -1.546886272 + 9.5 = 7.953113728
a = 7.953113728

Simplifying
a = 7.953113728

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {11.046886272, 7.953113728}
Solution
a = {11.046886272, 7.953113728}