3a-2(a-3)a-1=1

Solution for 3a-2(a-3)a-1=1 equation:

Simplifying
3a + -2(a + -3) * a + -1 = 1

Reorder the terms:
3a + -2(-3 + a) * a + -1 = 1

Reorder the terms for easier multiplication:
3a + -2a(-3 + a) + -1 = 1
3a + (-3 * -2a + a * -2a) + -1 = 1
3a + (6a + -2a2) + -1 = 1

Reorder the terms:
-1 + 3a + 6a + -2a2 = 1

Combine like terms: 3a + 6a = 9a
-1 + 9a + -2a2 = 1

Solving
-1 + 9a + -2a2 = 1

Solving for variable 'a'.

Reorder the terms:
-1 + -1 + 9a + -2a2 = 1 + -1

Combine like terms: -1 + -1 = -2
-2 + 9a + -2a2 = 1 + -1

Combine like terms: 1 + -1 = 0
-2 + 9a + -2a2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
1 + -4.5a + a2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4.5a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + -4.5a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4.5a + a2 = 0 + -1
-4.5a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4.5a + a2 = -1

The a term is -4.5a.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5a + 5.0625 + a2 = -1 + 5.0625

Reorder the terms:
5.0625 + -4.5a + a2 = -1 + 5.0625

Combine like terms: -1 + 5.0625 = 4.0625
5.0625 + -4.5a + a2 = 4.0625

Factor a perfect square on the left side:
(a + -2.25)(a + -2.25) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(a + -2.25) equal to 2.015564437 and -2.015564437.

Subproblem 1
a + -2.25 = 2.015564437

Simplifying
a + -2.25 = 2.015564437

Reorder the terms:
-2.25 + a = 2.015564437

Solving
-2.25 + a = 2.015564437

Solving for variable 'a'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + a = 2.015564437 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + a = 2.015564437 + 2.25
a = 2.015564437 + 2.25

Combine like terms: 2.015564437 + 2.25 = 4.265564437
a = 4.265564437

Simplifying
a = 4.265564437

Subproblem 2
a + -2.25 = -2.015564437

Simplifying
a + -2.25 = -2.015564437

Reorder the terms:
-2.25 + a = -2.015564437

Solving
-2.25 + a = -2.015564437

Solving for variable 'a'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + a = -2.015564437 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + a = -2.015564437 + 2.25
a = -2.015564437 + 2.25

Combine like terms: -2.015564437 + 2.25 = 0.234435563
a = 0.234435563

Simplifying
a = 0.234435563

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {4.265564437, 0.234435563}