(n+73)(n+77)=10000

Solution for (n+73)(n+77)=10000 equation:

Simplifying
(n + 73)(n + 77) = 10000

Reorder the terms:
(73 + n)(n + 77) = 10000

Reorder the terms:
(73 + n)(77 + n) = 10000

Multiply (73 + n) * (77 + n)
(73(77 + n) + n(77 + n)) = 10000
((77 * 73 + n * 73) + n(77 + n)) = 10000
((5621 + 73n) + n(77 + n)) = 10000
(5621 + 73n + (77 * n + n * n)) = 10000
(5621 + 73n + (77n + n2)) = 10000

Combine like terms: 73n + 77n = 150n
(5621 + 150n + n2) = 10000

Solving
5621 + 150n + n2 = 10000

Solving for variable 'n'.

Reorder the terms:
5621 + -10000 + 150n + n2 = 10000 + -10000

Combine like terms: 5621 + -10000 = -4379
-4379 + 150n + n2 = 10000 + -10000

Combine like terms: 10000 + -10000 = 0
-4379 + 150n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4379' to each side of the equation.
-4379 + 150n + 4379 + n2 = 0 + 4379

Reorder the terms:
-4379 + 4379 + 150n + n2 = 0 + 4379

Combine like terms: -4379 + 4379 = 0
0 + 150n + n2 = 0 + 4379
150n + n2 = 0 + 4379

Combine like terms: 0 + 4379 = 4379
150n + n2 = 4379

The n term is 150n.  Take half its coefficient (75).
Square it (5625) and add it to both sides.

Add '5625' to each side of the equation.
150n + 5625 + n2 = 4379 + 5625

Reorder the terms:
5625 + 150n + n2 = 4379 + 5625

Combine like terms: 4379 + 5625 = 10004
5625 + 150n + n2 = 10004

Factor a perfect square on the left side:
(n + 75)(n + 75) = 10004

Calculate the square root of the right side: 100.019998

Break this problem into two subproblems by setting 
(n + 75) equal to 100.019998 and -100.019998.

Subproblem 1
n + 75 = 100.019998

Simplifying
n + 75 = 100.019998

Reorder the terms:
75 + n = 100.019998

Solving
75 + n = 100.019998

Solving for variable 'n'.

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

Add '-75' to each side of the equation.
75 + -75 + n = 100.019998 + -75

Combine like terms: 75 + -75 = 0
0 + n = 100.019998 + -75
n = 100.019998 + -75

Combine like terms: 100.019998 + -75 = 25.019998
n = 25.019998

Simplifying
n = 25.019998

Subproblem 2
n + 75 = -100.019998

Simplifying
n + 75 = -100.019998

Reorder the terms:
75 + n = -100.019998

Solving
75 + n = -100.019998

Solving for variable 'n'.

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

Add '-75' to each side of the equation.
75 + -75 + n = -100.019998 + -75

Combine like terms: 75 + -75 = 0
0 + n = -100.019998 + -75
n = -100.019998 + -75

Combine like terms: -100.019998 + -75 = -175.019998
n = -175.019998

Simplifying
n = -175.019998

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {25.019998, -175.019998}