2t(t+3)+3=9

Solution for 2t(t+3)+3=9 equation:

Simplifying
2t(t + 3) + 3 = 9

Reorder the terms:
2t(3 + t) + 3 = 9
(3 * 2t + t * 2t) + 3 = 9
(6t + 2t2) + 3 = 9

Reorder the terms:
3 + 6t + 2t2 = 9

Solving
3 + 6t + 2t2 = 9

Solving for variable 't'.

Reorder the terms:
3 + -9 + 6t + 2t2 = 9 + -9

Combine like terms: 3 + -9 = -6
-6 + 6t + 2t2 = 9 + -9

Combine like terms: 9 + -9 = 0
-6 + 6t + 2t2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 3t + t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + 3t + t2)' equal to zero and attempt to solve:

Simplifying
-3 + 3t + t2 = 0

Solving
-3 + 3t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 3t + 3 + t2 = 0 + 3

Reorder the terms:
-3 + 3 + 3t + t2 = 0 + 3

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

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

The t term is 3t.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3t + 2.25 + t2 = 3 + 2.25

Reorder the terms:
2.25 + 3t + t2 = 3 + 2.25

Combine like terms: 3 + 2.25 = 5.25
2.25 + 3t + t2 = 5.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(t + 1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
t + 1.5 = 2.291287847

Simplifying
t + 1.5 = 2.291287847

Reorder the terms:
1.5 + t = 2.291287847

Solving
1.5 + t = 2.291287847

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = 2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = 2.291287847 + -1.5
t = 2.291287847 + -1.5

Combine like terms: 2.291287847 + -1.5 = 0.791287847
t = 0.791287847

Simplifying
t = 0.791287847

Subproblem 2
t + 1.5 = -2.291287847

Simplifying
t + 1.5 = -2.291287847

Reorder the terms:
1.5 + t = -2.291287847

Solving
1.5 + t = -2.291287847

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = -2.291287847 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = -2.291287847 + -1.5
t = -2.291287847 + -1.5

Combine like terms: -2.291287847 + -1.5 = -3.791287847
t = -3.791287847

Simplifying
t = -3.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.791287847, -3.791287847}
Solution
t = {0.791287847, -3.791287847}