10(3t+1)-12=t(t+1)

Solution for 10(3t+1)-12=t(t+1) equation:

Simplifying
10(3t + 1) + -12 = t(t + 1)

Reorder the terms:
10(1 + 3t) + -12 = t(t + 1)
(1 * 10 + 3t * 10) + -12 = t(t + 1)
(10 + 30t) + -12 = t(t + 1)

Reorder the terms:
10 + -12 + 30t = t(t + 1)

Combine like terms: 10 + -12 = -2
-2 + 30t = t(t + 1)

Reorder the terms:
-2 + 30t = t(1 + t)
-2 + 30t = (1 * t + t * t)
-2 + 30t = (1t + t2)

Solving
-2 + 30t = 1t + t2

Solving for variable 't'.

Combine like terms: 30t + -1t = 29t
-2 + 29t + -1t2 = 1t + t2 + -1t + -1t2

Reorder the terms:
-2 + 29t + -1t2 = 1t + -1t + t2 + -1t2

Combine like terms: 1t + -1t = 0
-2 + 29t + -1t2 = 0 + t2 + -1t2
-2 + 29t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-2 + 29t + -1t2 = 0

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

Divide each side by '-1'.
2 + -29t + t2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -29t + -2 + t2 = 0 + -2

Reorder the terms:
2 + -2 + -29t + t2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -29t + t2 = 0 + -2
-29t + t2 = 0 + -2

Combine like terms: 0 + -2 = -2
-29t + t2 = -2

The t term is -29t.  Take half its coefficient (-14.5).
Square it (210.25) and add it to both sides.

Add '210.25' to each side of the equation.
-29t + 210.25 + t2 = -2 + 210.25

Reorder the terms:
210.25 + -29t + t2 = -2 + 210.25

Combine like terms: -2 + 210.25 = 208.25
210.25 + -29t + t2 = 208.25

Factor a perfect square on the left side:
(t + -14.5)(t + -14.5) = 208.25

Calculate the square root of the right side: 14.43086969

Break this problem into two subproblems by setting 
(t + -14.5) equal to 14.43086969 and -14.43086969.

Subproblem 1
t + -14.5 = 14.43086969

Simplifying
t + -14.5 = 14.43086969

Reorder the terms:
-14.5 + t = 14.43086969

Solving
-14.5 + t = 14.43086969

Solving for variable 't'.

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

Add '14.5' to each side of the equation.
-14.5 + 14.5 + t = 14.43086969 + 14.5

Combine like terms: -14.5 + 14.5 = 0.0
0.0 + t = 14.43086969 + 14.5
t = 14.43086969 + 14.5

Combine like terms: 14.43086969 + 14.5 = 28.93086969
t = 28.93086969

Simplifying
t = 28.93086969

Subproblem 2
t + -14.5 = -14.43086969

Simplifying
t + -14.5 = -14.43086969

Reorder the terms:
-14.5 + t = -14.43086969

Solving
-14.5 + t = -14.43086969

Solving for variable 't'.

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

Add '14.5' to each side of the equation.
-14.5 + 14.5 + t = -14.43086969 + 14.5

Combine like terms: -14.5 + 14.5 = 0.0
0.0 + t = -14.43086969 + 14.5
t = -14.43086969 + 14.5

Combine like terms: -14.43086969 + 14.5 = 0.06913031
t = 0.06913031

Simplifying
t = 0.06913031

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {28.93086969, 0.06913031}