-10t^2+105t+30=75

Solution for -10t^2+105t+30=75 equation:

Simplifying
-10t2 + 105t + 30 = 75

Reorder the terms:
30 + 105t + -10t2 = 75

Solving
30 + 105t + -10t2 = 75

Solving for variable 't'.

Reorder the terms:
30 + -75 + 105t + -10t2 = 75 + -75

Combine like terms: 30 + -75 = -45
-45 + 105t + -10t2 = 75 + -75

Combine like terms: 75 + -75 = 0
-45 + 105t + -10t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-9 + 21t + -2t2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-9 + 21t + -2t2)' equal to zero and attempt to solve:

Simplifying
-9 + 21t + -2t2 = 0

Solving
-9 + 21t + -2t2 = 0

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

Divide each side by '-2'.
4.5 + -10.5t + t2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + -10.5t + -4.5 + t2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + -10.5t + t2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + -10.5t + t2 = 0 + -4.5
-10.5t + t2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
-10.5t + t2 = -4.5

The t term is -10.5t.  Take half its coefficient (-5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
-10.5t + 27.5625 + t2 = -4.5 + 27.5625

Reorder the terms:
27.5625 + -10.5t + t2 = -4.5 + 27.5625

Combine like terms: -4.5 + 27.5625 = 23.0625
27.5625 + -10.5t + t2 = 23.0625

Factor a perfect square on the left side:
(t + -5.25)(t + -5.25) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(t + -5.25) equal to 4.802343178 and -4.802343178.

Subproblem 1
t + -5.25 = 4.802343178

Simplifying
t + -5.25 = 4.802343178

Reorder the terms:
-5.25 + t = 4.802343178

Solving
-5.25 + t = 4.802343178

Solving for variable 't'.

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

Add '5.25' to each side of the equation.
-5.25 + 5.25 + t = 4.802343178 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + t = 4.802343178 + 5.25
t = 4.802343178 + 5.25

Combine like terms: 4.802343178 + 5.25 = 10.052343178
t = 10.052343178

Simplifying
t = 10.052343178

Subproblem 2
t + -5.25 = -4.802343178

Simplifying
t + -5.25 = -4.802343178

Reorder the terms:
-5.25 + t = -4.802343178

Solving
-5.25 + t = -4.802343178

Solving for variable 't'.

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

Add '5.25' to each side of the equation.
-5.25 + 5.25 + t = -4.802343178 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + t = -4.802343178 + 5.25
t = -4.802343178 + 5.25

Combine like terms: -4.802343178 + 5.25 = 0.447656822
t = 0.447656822

Simplifying
t = 0.447656822

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {10.052343178, 0.447656822}
Solution
t = {10.052343178, 0.447656822}