0.5t(t-12)=1.5t-3

Solution for 0.5t(t-12)=1.5t-3 equation:

Simplifying
0.5t(t + -12) = 1.5t + -3

Reorder the terms:
0.5t(-12 + t) = 1.5t + -3
(-12 * 0.5t + t * 0.5t) = 1.5t + -3
(-6t + 0.5t2) = 1.5t + -3

Reorder the terms:
-6t + 0.5t2 = -3 + 1.5t

Solving
-6t + 0.5t2 = -3 + 1.5t

Solving for variable 't'.

Reorder the terms:
3 + -6t + -1.5t + 0.5t2 = -3 + 1.5t + 3 + -1.5t

Combine like terms: -6t + -1.5t = -7.5t
3 + -7.5t + 0.5t2 = -3 + 1.5t + 3 + -1.5t

Reorder the terms:
3 + -7.5t + 0.5t2 = -3 + 3 + 1.5t + -1.5t

Combine like terms: -3 + 3 = 0
3 + -7.5t + 0.5t2 = 0 + 1.5t + -1.5t
3 + -7.5t + 0.5t2 = 1.5t + -1.5t

Combine like terms: 1.5t + -1.5t = 0.0
3 + -7.5t + 0.5t2 = 0.0

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

Divide each side by '0.5'.
6 + -15t + t2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -15t + -6 + t2 = 0 + -6

Reorder the terms:
6 + -6 + -15t + t2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -15t + t2 = 0 + -6
-15t + t2 = 0 + -6

Combine like terms: 0 + -6 = -6
-15t + t2 = -6

The t term is -15t.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15t + 56.25 + t2 = -6 + 56.25

Reorder the terms:
56.25 + -15t + t2 = -6 + 56.25

Combine like terms: -6 + 56.25 = 50.25
56.25 + -15t + t2 = 50.25

Factor a perfect square on the left side:
(t + -7.5)(t + -7.5) = 50.25

Calculate the square root of the right side: 7.088723439

Break this problem into two subproblems by setting 
(t + -7.5) equal to 7.088723439 and -7.088723439.

Subproblem 1
t + -7.5 = 7.088723439

Simplifying
t + -7.5 = 7.088723439

Reorder the terms:
-7.5 + t = 7.088723439

Solving
-7.5 + t = 7.088723439

Solving for variable 't'.

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

Add '7.5' to each side of the equation.
-7.5 + 7.5 + t = 7.088723439 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + t = 7.088723439 + 7.5
t = 7.088723439 + 7.5

Combine like terms: 7.088723439 + 7.5 = 14.588723439
t = 14.588723439

Simplifying
t = 14.588723439

Subproblem 2
t + -7.5 = -7.088723439

Simplifying
t + -7.5 = -7.088723439

Reorder the terms:
-7.5 + t = -7.088723439

Solving
-7.5 + t = -7.088723439

Solving for variable 't'.

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

Add '7.5' to each side of the equation.
-7.5 + 7.5 + t = -7.088723439 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + t = -7.088723439 + 7.5
t = -7.088723439 + 7.5

Combine like terms: -7.088723439 + 7.5 = 0.411276561
t = 0.411276561

Simplifying
t = 0.411276561

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {14.588723439, 0.411276561}