k+k(k-45)+(k-45)+(k-45)=540

Solution for k+k(k-45)+(k-45)+(k-45)=540 equation:

Simplifying
k + k(k + -45) + (k + -45) + (k + -45) = 540

Reorder the terms:
k + k(-45 + k) + (k + -45) + (k + -45) = 540
k + (-45 * k + k * k) + (k + -45) + (k + -45) = 540
k + (-45k + k2) + (k + -45) + (k + -45) = 540

Reorder the terms:
k + -45k + k2 + (-45 + k) + (k + -45) = 540

Remove parenthesis around (-45 + k)
k + -45k + k2 + -45 + k + (k + -45) = 540

Reorder the terms:
k + -45k + k2 + -45 + k + (-45 + k) = 540

Remove parenthesis around (-45 + k)
k + -45k + k2 + -45 + k + -45 + k = 540

Reorder the terms:
-45 + -45 + k + -45k + k + k + k2 = 540

Combine like terms: -45 + -45 = -90
-90 + k + -45k + k + k + k2 = 540

Combine like terms: k + -45k = -44k
-90 + -44k + k + k + k2 = 540

Combine like terms: -44k + k = -43k
-90 + -43k + k + k2 = 540

Combine like terms: -43k + k = -42k
-90 + -42k + k2 = 540

Solving
-90 + -42k + k2 = 540

Solving for variable 'k'.

Reorder the terms:
-90 + -540 + -42k + k2 = 540 + -540

Combine like terms: -90 + -540 = -630
-630 + -42k + k2 = 540 + -540

Combine like terms: 540 + -540 = 0
-630 + -42k + k2 = 0

Begin completing the square.

Move the constant term to the right:

Add '630' to each side of the equation.
-630 + -42k + 630 + k2 = 0 + 630

Reorder the terms:
-630 + 630 + -42k + k2 = 0 + 630

Combine like terms: -630 + 630 = 0
0 + -42k + k2 = 0 + 630
-42k + k2 = 0 + 630

Combine like terms: 0 + 630 = 630
-42k + k2 = 630

The k term is -42k.  Take half its coefficient (-21).
Square it (441) and add it to both sides.

Add '441' to each side of the equation.
-42k + 441 + k2 = 630 + 441

Reorder the terms:
441 + -42k + k2 = 630 + 441

Combine like terms: 630 + 441 = 1071
441 + -42k + k2 = 1071

Factor a perfect square on the left side:
(k + -21)(k + -21) = 1071

Calculate the square root of the right side: 32.726136344

Break this problem into two subproblems by setting 
(k + -21) equal to 32.726136344 and -32.726136344.

Subproblem 1
k + -21 = 32.726136344

Simplifying
k + -21 = 32.726136344

Reorder the terms:
-21 + k = 32.726136344

Solving
-21 + k = 32.726136344

Solving for variable 'k'.

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

Add '21' to each side of the equation.
-21 + 21 + k = 32.726136344 + 21

Combine like terms: -21 + 21 = 0
0 + k = 32.726136344 + 21
k = 32.726136344 + 21

Combine like terms: 32.726136344 + 21 = 53.726136344
k = 53.726136344

Simplifying
k = 53.726136344

Subproblem 2
k + -21 = -32.726136344

Simplifying
k + -21 = -32.726136344

Reorder the terms:
-21 + k = -32.726136344

Solving
-21 + k = -32.726136344

Solving for variable 'k'.

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

Add '21' to each side of the equation.
-21 + 21 + k = -32.726136344 + 21

Combine like terms: -21 + 21 = 0
0 + k = -32.726136344 + 21
k = -32.726136344 + 21

Combine like terms: -32.726136344 + 21 = -11.726136344
k = -11.726136344

Simplifying
k = -11.726136344

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {53.726136344, -11.726136344}