(n+1)(n-4)=52

Solution for (n+1)(n-4)=52 equation:

Simplifying
(n + 1)(n + -4) = 52

Reorder the terms:
(1 + n)(n + -4) = 52

Reorder the terms:
(1 + n)(-4 + n) = 52

Multiply (1 + n) * (-4 + n)
(1(-4 + n) + n(-4 + n)) = 52
((-4 * 1 + n * 1) + n(-4 + n)) = 52
((-4 + 1n) + n(-4 + n)) = 52
(-4 + 1n + (-4 * n + n * n)) = 52
(-4 + 1n + (-4n + n2)) = 52

Combine like terms: 1n + -4n = -3n
(-4 + -3n + n2) = 52

Solving
-4 + -3n + n2 = 52

Solving for variable 'n'.

Reorder the terms:
-4 + -52 + -3n + n2 = 52 + -52

Combine like terms: -4 + -52 = -56
-56 + -3n + n2 = 52 + -52

Combine like terms: 52 + -52 = 0
-56 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '56' to each side of the equation.
-56 + -3n + 56 + n2 = 0 + 56

Reorder the terms:
-56 + 56 + -3n + n2 = 0 + 56

Combine like terms: -56 + 56 = 0
0 + -3n + n2 = 0 + 56
-3n + n2 = 0 + 56

Combine like terms: 0 + 56 = 56
-3n + n2 = 56

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 56 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 56 + 2.25

Combine like terms: 56 + 2.25 = 58.25
2.25 + -3n + n2 = 58.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 58.25

Calculate the square root of the right side: 7.632168761

Break this problem into two subproblems by setting 
(n + -1.5) equal to 7.632168761 and -7.632168761.

Subproblem 1
n + -1.5 = 7.632168761

Simplifying
n + -1.5 = 7.632168761

Reorder the terms:
-1.5 + n = 7.632168761

Solving
-1.5 + n = 7.632168761

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 7.632168761 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 7.632168761 + 1.5
n = 7.632168761 + 1.5

Combine like terms: 7.632168761 + 1.5 = 9.132168761
n = 9.132168761

Simplifying
n = 9.132168761

Subproblem 2
n + -1.5 = -7.632168761

Simplifying
n + -1.5 = -7.632168761

Reorder the terms:
-1.5 + n = -7.632168761

Solving
-1.5 + n = -7.632168761

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -7.632168761 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -7.632168761 + 1.5
n = -7.632168761 + 1.5

Combine like terms: -7.632168761 + 1.5 = -6.132168761
n = -6.132168761

Simplifying
n = -6.132168761

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9.132168761, -6.132168761}