7t(t+3)=35

Solution for 7t(t+3)=35 equation:

Simplifying
7t(t + 3) = 35

Reorder the terms:
7t(3 + t) = 35
(3 * 7t + t * 7t) = 35
(21t + 7t2) = 35

Solving
21t + 7t2 = 35

Solving for variable 't'.

Reorder the terms:
-35 + 21t + 7t2 = 35 + -35

Combine like terms: 35 + -35 = 0
-35 + 21t + 7t2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-5 + 3t + t2) = 0

Ignore the factor 7.
Subproblem 1
Set the factor '(-5 + 3t + t2)' equal to zero and attempt to solve:

Simplifying
-5 + 3t + t2 = 0

Solving
-5 + 3t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 3t + 5 + t2 = 0 + 5

Reorder the terms:
-5 + 5 + 3t + t2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 3t + t2 = 0 + 5
3t + t2 = 0 + 5

Combine like terms: 0 + 5 = 5
3t + t2 = 5

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 = 5 + 2.25

Reorder the terms:
2.25 + 3t + t2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3t + t2 = 7.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(t + 1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1
t + 1.5 = 2.692582404

Simplifying
t + 1.5 = 2.692582404

Reorder the terms:
1.5 + t = 2.692582404

Solving
1.5 + t = 2.692582404

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 = 2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = 2.692582404 + -1.5
t = 2.692582404 + -1.5

Combine like terms: 2.692582404 + -1.5 = 1.192582404
t = 1.192582404

Simplifying
t = 1.192582404

Subproblem 2
t + 1.5 = -2.692582404

Simplifying
t + 1.5 = -2.692582404

Reorder the terms:
1.5 + t = -2.692582404

Solving
1.5 + t = -2.692582404

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 = -2.692582404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = -2.692582404 + -1.5
t = -2.692582404 + -1.5

Combine like terms: -2.692582404 + -1.5 = -4.192582404
t = -4.192582404

Simplifying
t = -4.192582404

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.192582404, -4.192582404}
Solution
t = {1.192582404, -4.192582404}