3t(t+2)-3t=19

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Solution for 3t(t+2)-3t=19 equation: 


Simplifying
3t(t + 2) + -3t = 19

Reorder the terms:
3t(2 + t) + -3t = 19
(2 * 3t + t * 3t) + -3t = 19
(6t + 3t2) + -3t = 19

Reorder the terms:
6t + -3t + 3t2 = 19

Combine like terms: 6t + -3t = 3t
3t + 3t2 = 19

Solving
3t + 3t2 = 19

Solving for variable 't'.

Reorder the terms:
-19 + 3t + 3t2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + 3t + 3t2 = 0

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

Divide each side by '3'.
-6.333333333 + t + t2 = 0

Move the constant term to the right:

Add '6.333333333' to each side of the equation.
-6.333333333 + t + 6.333333333 + t2 = 0 + 6.333333333

Reorder the terms:
-6.333333333 + 6.333333333 + t + t2 = 0 + 6.333333333

Combine like terms: -6.333333333 + 6.333333333 = 0.000000000
0.000000000 + t + t2 = 0 + 6.333333333
t + t2 = 0 + 6.333333333

Combine like terms: 0 + 6.333333333 = 6.333333333
t + t2 = 6.333333333

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + t2 = 6.333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + t2 = 6.333333333 + 0.25

Combine like terms: 6.333333333 + 0.25 = 6.583333333
1.25 + t2 = 6.583333333

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 6.583333333

Calculate the square root of the right side: 2.56580072

Break this problem into two subproblems by setting 
(t + 0.5) equal to 2.56580072 and -2.56580072.


Subproblem 1
t + 0.5 = 2.56580072

Simplifying
t + 0.5 = 2.56580072

Reorder the terms:
0.5 + t = 2.56580072

Solving
0.5 + t = 2.56580072

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 2.56580072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 2.56580072 + -0.5
t = 2.56580072 + -0.5

Combine like terms: 2.56580072 + -0.5 = 2.06580072
t = 2.06580072

Simplifying
t = 2.06580072


Subproblem 2
t + 0.5 = -2.56580072

Simplifying
t + 0.5 = -2.56580072

Reorder the terms:
0.5 + t = -2.56580072

Solving
0.5 + t = -2.56580072

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -2.56580072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -2.56580072 + -0.5
t = -2.56580072 + -0.5

Combine like terms: -2.56580072 + -0.5 = -3.06580072
t = -3.06580072

Simplifying
t = -3.06580072


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.06580072, -3.06580072}

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