(n-3)(n-3)=19

Solution for (n-3)(n-3)=19 equation:

Simplifying
(n + -3)(n + -3) = 19

Reorder the terms:
(-3 + n)(n + -3) = 19

Reorder the terms:
(-3 + n)(-3 + n) = 19

Multiply (-3 + n) * (-3 + n)
(-3(-3 + n) + n(-3 + n)) = 19
((-3 * -3 + n * -3) + n(-3 + n)) = 19
((9 + -3n) + n(-3 + n)) = 19
(9 + -3n + (-3 * n + n * n)) = 19
(9 + -3n + (-3n + n2)) = 19

Combine like terms: -3n + -3n = -6n
(9 + -6n + n2) = 19

Solving
9 + -6n + n2 = 19

Solving for variable 'n'.

Reorder the terms:
9 + -19 + -6n + n2 = 19 + -19

Combine like terms: 9 + -19 = -10
-10 + -6n + n2 = 19 + -19

Combine like terms: 19 + -19 = 0
-10 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -6n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + -6n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -6n + n2 = 0 + 10
-6n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
-6n + n2 = 10

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 10 + 9

Reorder the terms:
9 + -6n + n2 = 10 + 9

Combine like terms: 10 + 9 = 19
9 + -6n + n2 = 19

Factor a perfect square on the left side:
(n + -3)(n + -3) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(n + -3) equal to 4.358898944 and -4.358898944.

Subproblem 1
n + -3 = 4.358898944

Simplifying
n + -3 = 4.358898944

Reorder the terms:
-3 + n = 4.358898944

Solving
-3 + n = 4.358898944

Solving for variable 'n'.

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

Add '3' to each side of the equation.
-3 + 3 + n = 4.358898944 + 3

Combine like terms: -3 + 3 = 0
0 + n = 4.358898944 + 3
n = 4.358898944 + 3

Combine like terms: 4.358898944 + 3 = 7.358898944
n = 7.358898944

Simplifying
n = 7.358898944

Subproblem 2
n + -3 = -4.358898944

Simplifying
n + -3 = -4.358898944

Reorder the terms:
-3 + n = -4.358898944

Solving
-3 + n = -4.358898944

Solving for variable 'n'.

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

Add '3' to each side of the equation.
-3 + 3 + n = -4.358898944 + 3

Combine like terms: -3 + 3 = 0
0 + n = -4.358898944 + 3
n = -4.358898944 + 3

Combine like terms: -4.358898944 + 3 = -1.358898944
n = -1.358898944

Simplifying
n = -1.358898944

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.358898944, -1.358898944}