-5t^2+105t+30=75

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Solution for -5t^2+105t+30=75 equation: 


Simplifying
-5t2 + 105t + 30 = 75

Reorder the terms:
30 + 105t + -5t2 = 75

Solving
30 + 105t + -5t2 = 75

Solving for variable 't'.

Reorder the terms:
30 + -75 + 105t + -5t2 = 75 + -75

Combine like terms: 30 + -75 = -45
-45 + 105t + -5t2 = 75 + -75

Combine like terms: 75 + -75 = 0
-45 + 105t + -5t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-9 + 21t + -1t2) = 0

Ignore the factor 5.

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

Simplifying
-9 + 21t + -1t2 = 0

Solving
-9 + 21t + -1t2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
9 + -21t + t2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -21t + -9 + t2 = 0 + -9

Reorder the terms:
9 + -9 + -21t + t2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -21t + t2 = 0 + -9
-21t + t2 = 0 + -9

Combine like terms: 0 + -9 = -9
-21t + t2 = -9

The t term is -21t.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21t + 110.25 + t2 = -9 + 110.25

Reorder the terms:
110.25 + -21t + t2 = -9 + 110.25

Combine like terms: -9 + 110.25 = 101.25
110.25 + -21t + t2 = 101.25

Factor a perfect square on the left side:
(t + -10.5)(t + -10.5) = 101.25

Calculate the square root of the right side: 10.062305899

Break this problem into two subproblems by setting 
(t + -10.5) equal to 10.062305899 and -10.062305899.


Subproblem 1
t + -10.5 = 10.062305899

Simplifying
t + -10.5 = 10.062305899

Reorder the terms:
-10.5 + t = 10.062305899

Solving
-10.5 + t = 10.062305899

Solving for variable 't'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + t = 10.062305899 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + t = 10.062305899 + 10.5
t = 10.062305899 + 10.5

Combine like terms: 10.062305899 + 10.5 = 20.562305899
t = 20.562305899

Simplifying
t = 20.562305899


Subproblem 2
t + -10.5 = -10.062305899

Simplifying
t + -10.5 = -10.062305899

Reorder the terms:
-10.5 + t = -10.062305899

Solving
-10.5 + t = -10.062305899

Solving for variable 't'.

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

Add '10.5' to each side of the equation.
-10.5 + 10.5 + t = -10.062305899 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + t = -10.062305899 + 10.5
t = -10.062305899 + 10.5

Combine like terms: -10.062305899 + 10.5 = 0.437694101
t = 0.437694101

Simplifying
t = 0.437694101


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {20.562305899, 0.437694101}
Solution
t = {20.562305899, 0.437694101}

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