12m^2-27m-75=0

Solution for 12m^2-27m-75=0 equation:

Simplifying
12m2 + -27m + -75 = 0

Reorder the terms:
-75 + -27m + 12m2 = 0

Solving
-75 + -27m + 12m2 = 0

Solving for variable 'm'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-25 + -9m + 4m2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-25 + -9m + 4m2)' equal to zero and attempt to solve:

Simplifying
-25 + -9m + 4m2 = 0

Solving
-25 + -9m + 4m2 = 0

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

Divide each side by '4'.
-6.25 + -2.25m + m2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + -2.25m + 6.25 + m2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + -2.25m + m2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + -2.25m + m2 = 0 + 6.25
-2.25m + m2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
-2.25m + m2 = 6.25

The m term is -2.25m.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25m + 1.265625 + m2 = 6.25 + 1.265625

Reorder the terms:
1.265625 + -2.25m + m2 = 6.25 + 1.265625

Combine like terms: 6.25 + 1.265625 = 7.515625
1.265625 + -2.25m + m2 = 7.515625

Factor a perfect square on the left side:
(m + -1.125)(m + -1.125) = 7.515625

Calculate the square root of the right side: 2.741464025

Break this problem into two subproblems by setting 
(m + -1.125) equal to 2.741464025 and -2.741464025.

Subproblem 1
m + -1.125 = 2.741464025

Simplifying
m + -1.125 = 2.741464025

Reorder the terms:
-1.125 + m = 2.741464025

Solving
-1.125 + m = 2.741464025

Solving for variable 'm'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + m = 2.741464025 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + m = 2.741464025 + 1.125
m = 2.741464025 + 1.125

Combine like terms: 2.741464025 + 1.125 = 3.866464025
m = 3.866464025

Simplifying
m = 3.866464025

Subproblem 2
m + -1.125 = -2.741464025

Simplifying
m + -1.125 = -2.741464025

Reorder the terms:
-1.125 + m = -2.741464025

Solving
-1.125 + m = -2.741464025

Solving for variable 'm'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + m = -2.741464025 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + m = -2.741464025 + 1.125
m = -2.741464025 + 1.125

Combine like terms: -2.741464025 + 1.125 = -1.616464025
m = -1.616464025

Simplifying
m = -1.616464025

Solution
The solution to the problem is based on the solutions
from the subproblems.
m = {3.866464025, -1.616464025}
Solution
m = {3.866464025, -1.616464025}