-3j^2-54j=-75

Solution for -3j^2-54j=-75 equation:

Simplifying
-3j2 + -54j = -75

Reorder the terms:
-54j + -3j2 = -75

Solving
-54j + -3j2 = -75

Solving for variable 'j'.

Reorder the terms:
75 + -54j + -3j2 = -75 + 75

Combine like terms: -75 + 75 = 0
75 + -54j + -3j2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(25 + -18j + -1j2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(25 + -18j + -1j2)' equal to zero and attempt to solve:

Simplifying
25 + -18j + -1j2 = 0

Solving
25 + -18j + -1j2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-25 + 18j + j2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 18j + 25 + j2 = 0 + 25

Reorder the terms:
-25 + 25 + 18j + j2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 18j + j2 = 0 + 25
18j + j2 = 0 + 25

Combine like terms: 0 + 25 = 25
18j + j2 = 25

The j term is 18j.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18j + 81 + j2 = 25 + 81

Reorder the terms:
81 + 18j + j2 = 25 + 81

Combine like terms: 25 + 81 = 106
81 + 18j + j2 = 106

Factor a perfect square on the left side:
(j + 9)(j + 9) = 106

Calculate the square root of the right side: 10.295630141

Break this problem into two subproblems by setting 
(j + 9) equal to 10.295630141 and -10.295630141.

Subproblem 1
j + 9 = 10.295630141

Simplifying
j + 9 = 10.295630141

Reorder the terms:
9 + j = 10.295630141

Solving
9 + j = 10.295630141

Solving for variable 'j'.

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

Add '-9' to each side of the equation.
9 + -9 + j = 10.295630141 + -9

Combine like terms: 9 + -9 = 0
0 + j = 10.295630141 + -9
j = 10.295630141 + -9

Combine like terms: 10.295630141 + -9 = 1.295630141
j = 1.295630141

Simplifying
j = 1.295630141

Subproblem 2
j + 9 = -10.295630141

Simplifying
j + 9 = -10.295630141

Reorder the terms:
9 + j = -10.295630141

Solving
9 + j = -10.295630141

Solving for variable 'j'.

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

Add '-9' to each side of the equation.
9 + -9 + j = -10.295630141 + -9

Combine like terms: 9 + -9 = 0
0 + j = -10.295630141 + -9
j = -10.295630141 + -9

Combine like terms: -10.295630141 + -9 = -19.295630141
j = -19.295630141

Simplifying
j = -19.295630141

Solution
The solution to the problem is based on the solutions
from the subproblems.
j = {1.295630141, -19.295630141}
Solution
j = {1.295630141, -19.295630141}