-7t(2t+3)=-105

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Solution for -7t(2t+3)=-105 equation: 


Simplifying
-7t(2t + 3) = -105

Reorder the terms:
-7t(3 + 2t) = -105
(3 * -7t + 2t * -7t) = -105
(-21t + -14t2) = -105

Solving
-21t + -14t2 = -105

Solving for variable 't'.

Reorder the terms:
105 + -21t + -14t2 = -105 + 105

Combine like terms: -105 + 105 = 0
105 + -21t + -14t2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(15 + -3t + -2t2) = 0

Ignore the factor 7.

Subproblem 1
Set the factor '(15 + -3t + -2t2)' equal to zero and attempt to solve:

Simplifying
15 + -3t + -2t2 = 0

Solving
15 + -3t + -2t2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-7.5 + 1.5t + t2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 1.5t + 7.5 + t2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 1.5t + t2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 1.5t + t2 = 0 + 7.5
1.5t + t2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
1.5t + t2 = 7.5

The t term is 1.5t.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5t + 0.5625 + t2 = 7.5 + 0.5625

Reorder the terms:
0.5625 + 1.5t + t2 = 7.5 + 0.5625

Combine like terms: 7.5 + 0.5625 = 8.0625
0.5625 + 1.5t + t2 = 8.0625

Factor a perfect square on the left side:
(t + 0.75)(t + 0.75) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(t + 0.75) equal to 2.839454173 and -2.839454173.


Subproblem 1
t + 0.75 = 2.839454173

Simplifying
t + 0.75 = 2.839454173

Reorder the terms:
0.75 + t = 2.839454173

Solving
0.75 + t = 2.839454173

Solving for variable 't'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + t = 2.839454173 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = 2.839454173 + -0.75
t = 2.839454173 + -0.75

Combine like terms: 2.839454173 + -0.75 = 2.089454173
t = 2.089454173

Simplifying
t = 2.089454173


Subproblem 2
t + 0.75 = -2.839454173

Simplifying
t + 0.75 = -2.839454173

Reorder the terms:
0.75 + t = -2.839454173

Solving
0.75 + t = -2.839454173

Solving for variable 't'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + t = -2.839454173 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + t = -2.839454173 + -0.75
t = -2.839454173 + -0.75

Combine like terms: -2.839454173 + -0.75 = -3.589454173
t = -3.589454173

Simplifying
t = -3.589454173


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.089454173, -3.589454173}
Solution
t = {2.089454173, -3.589454173}

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