3(3t+1)-8=t(t+4)

Solution for 3(3t+1)-8=t(t+4) equation:

Simplifying
3(3t + 1) + -8 = t(t + 4)

Reorder the terms:
3(1 + 3t) + -8 = t(t + 4)
(1 * 3 + 3t * 3) + -8 = t(t + 4)
(3 + 9t) + -8 = t(t + 4)

Reorder the terms:
3 + -8 + 9t = t(t + 4)

Combine like terms: 3 + -8 = -5
-5 + 9t = t(t + 4)

Reorder the terms:
-5 + 9t = t(4 + t)
-5 + 9t = (4 * t + t * t)
-5 + 9t = (4t + t2)

Solving
-5 + 9t = 4t + t2

Solving for variable 't'.

Combine like terms: 9t + -4t = 5t
-5 + 5t + -1t2 = 4t + t2 + -4t + -1t2

Reorder the terms:
-5 + 5t + -1t2 = 4t + -4t + t2 + -1t2

Combine like terms: 4t + -4t = 0
-5 + 5t + -1t2 = 0 + t2 + -1t2
-5 + 5t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-5 + 5t + -1t2 = 0

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

Divide each side by '-1'.
5 + -5t + t2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -5t + -5 + t2 = 0 + -5

Reorder the terms:
5 + -5 + -5t + t2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -5t + t2 = 0 + -5
-5t + t2 = 0 + -5

Combine like terms: 0 + -5 = -5
-5t + t2 = -5

The t term is -5t.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5t + 6.25 + t2 = -5 + 6.25

Reorder the terms:
6.25 + -5t + t2 = -5 + 6.25

Combine like terms: -5 + 6.25 = 1.25
6.25 + -5t + t2 = 1.25

Factor a perfect square on the left side:
(t + -2.5)(t + -2.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(t + -2.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
t + -2.5 = 1.118033989

Simplifying
t + -2.5 = 1.118033989

Reorder the terms:
-2.5 + t = 1.118033989

Solving
-2.5 + t = 1.118033989

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = 1.118033989 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = 1.118033989 + 2.5
t = 1.118033989 + 2.5

Combine like terms: 1.118033989 + 2.5 = 3.618033989
t = 3.618033989

Simplifying
t = 3.618033989

Subproblem 2
t + -2.5 = -1.118033989

Simplifying
t + -2.5 = -1.118033989

Reorder the terms:
-2.5 + t = -1.118033989

Solving
-2.5 + t = -1.118033989

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = -1.118033989 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = -1.118033989 + 2.5
t = -1.118033989 + 2.5

Combine like terms: -1.118033989 + 2.5 = 1.381966011
t = 1.381966011

Simplifying
t = 1.381966011

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.618033989, 1.381966011}