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

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

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

Reorder the terms:
a(-2 + a) = (a + 3) + -1(a + 1)
(-2 * a + a * a) = (a + 3) + -1(a + 1)
(-2a + a2) = (a + 3) + -1(a + 1)

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

Remove parenthesis around (3 + a)
-2a + a2 = 3 + a + -1(a + 1)

Reorder the terms:
-2a + a2 = 3 + a + -1(1 + a)
-2a + a2 = 3 + a + (1 * -1 + a * -1)
-2a + a2 = 3 + a + (-1 + -1a)

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

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

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

Solving
-2a + a2 = 2

Solving for variable 'a'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2a + 2 + a2 = 0 + 2

Reorder the terms:
-2 + 2 + -2a + a2 = 0 + 2

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

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

The a term is -2a.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2a + 1 + a2 = 2 + 1

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

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

Factor a perfect square on the left side:
(a + -1)(a + -1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(a + -1) equal to 1.732050808 and -1.732050808.

Subproblem 1
a + -1 = 1.732050808

Simplifying
a + -1 = 1.732050808

Reorder the terms:
-1 + a = 1.732050808

Solving
-1 + a = 1.732050808

Solving for variable 'a'.

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

Add '1' to each side of the equation.
-1 + 1 + a = 1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + a = 1.732050808 + 1
a = 1.732050808 + 1

Combine like terms: 1.732050808 + 1 = 2.732050808
a = 2.732050808

Simplifying
a = 2.732050808

Subproblem 2
a + -1 = -1.732050808

Simplifying
a + -1 = -1.732050808

Reorder the terms:
-1 + a = -1.732050808

Solving
-1 + a = -1.732050808

Solving for variable 'a'.

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

Add '1' to each side of the equation.
-1 + 1 + a = -1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + a = -1.732050808 + 1
a = -1.732050808 + 1

Combine like terms: -1.732050808 + 1 = -0.732050808
a = -0.732050808

Simplifying
a = -0.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.732050808, -0.732050808}