(u-1)(u+3)=1

Solution for (u-1)(u+3)=1 equation:

Simplifying
(u + -1)(u + 3) = 1

Reorder the terms:
(-1 + u)(u + 3) = 1

Reorder the terms:
(-1 + u)(3 + u) = 1

Multiply (-1 + u) * (3 + u)
(-1(3 + u) + u(3 + u)) = 1
((3 * -1 + u * -1) + u(3 + u)) = 1
((-3 + -1u) + u(3 + u)) = 1
(-3 + -1u + (3 * u + u * u)) = 1
(-3 + -1u + (3u + u2)) = 1

Combine like terms: -1u + 3u = 2u
(-3 + 2u + u2) = 1

Solving
-3 + 2u + u2 = 1

Solving for variable 'u'.

Reorder the terms:
-3 + -1 + 2u + u2 = 1 + -1

Combine like terms: -3 + -1 = -4
-4 + 2u + u2 = 1 + -1

Combine like terms: 1 + -1 = 0
-4 + 2u + u2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2u + 4 + u2 = 0 + 4

Reorder the terms:
-4 + 4 + 2u + u2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2u + u2 = 0 + 4
2u + u2 = 0 + 4

Combine like terms: 0 + 4 = 4
2u + u2 = 4

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

Add '1' to each side of the equation.
2u + 1 + u2 = 4 + 1

Reorder the terms:
1 + 2u + u2 = 4 + 1

Combine like terms: 4 + 1 = 5
1 + 2u + u2 = 5

Factor a perfect square on the left side:
(u + 1)(u + 1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(u + 1) equal to 2.236067978 and -2.236067978.

Subproblem 1
u + 1 = 2.236067978

Simplifying
u + 1 = 2.236067978

Reorder the terms:
1 + u = 2.236067978

Solving
1 + u = 2.236067978

Solving for variable 'u'.

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

Add '-1' to each side of the equation.
1 + -1 + u = 2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + u = 2.236067978 + -1
u = 2.236067978 + -1

Combine like terms: 2.236067978 + -1 = 1.236067978
u = 1.236067978

Simplifying
u = 1.236067978

Subproblem 2
u + 1 = -2.236067978

Simplifying
u + 1 = -2.236067978

Reorder the terms:
1 + u = -2.236067978

Solving
1 + u = -2.236067978

Solving for variable 'u'.

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

Add '-1' to each side of the equation.
1 + -1 + u = -2.236067978 + -1

Combine like terms: 1 + -1 = 0
0 + u = -2.236067978 + -1
u = -2.236067978 + -1

Combine like terms: -2.236067978 + -1 = -3.236067978
u = -3.236067978

Simplifying
u = -3.236067978

Solution
The solution to the problem is based on the solutions
from the subproblems.
u = {1.236067978, -3.236067978}