-120=45t-5t^2

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Solution for -120=45t-5t^2 equation: 


Simplifying
-120 = 45t + -5t2

Solving
-120 = 45t + -5t2

Solving for variable 't'.

Reorder the terms:
-120 + -45t + 5t2 = 45t + -45t + -5t2 + 5t2

Combine like terms: 45t + -45t = 0
-120 + -45t + 5t2 = 0 + -5t2 + 5t2
-120 + -45t + 5t2 = -5t2 + 5t2

Combine like terms: -5t2 + 5t2 = 0
-120 + -45t + 5t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-24 + -9t + t2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-24 + -9t + t2)' equal to zero and attempt to solve:

Simplifying
-24 + -9t + t2 = 0

Solving
-24 + -9t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -9t + 24 + t2 = 0 + 24

Reorder the terms:
-24 + 24 + -9t + t2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -9t + t2 = 0 + 24
-9t + t2 = 0 + 24

Combine like terms: 0 + 24 = 24
-9t + t2 = 24

The t term is -9t.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9t + 20.25 + t2 = 24 + 20.25

Reorder the terms:
20.25 + -9t + t2 = 24 + 20.25

Combine like terms: 24 + 20.25 = 44.25
20.25 + -9t + t2 = 44.25

Factor a perfect square on the left side:
(t + -4.5)(t + -4.5) = 44.25

Calculate the square root of the right side: 6.652067348

Break this problem into two subproblems by setting 
(t + -4.5) equal to 6.652067348 and -6.652067348.


Subproblem 1
t + -4.5 = 6.652067348

Simplifying
t + -4.5 = 6.652067348

Reorder the terms:
-4.5 + t = 6.652067348

Solving
-4.5 + t = 6.652067348

Solving for variable 't'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + t = 6.652067348 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + t = 6.652067348 + 4.5
t = 6.652067348 + 4.5

Combine like terms: 6.652067348 + 4.5 = 11.152067348
t = 11.152067348

Simplifying
t = 11.152067348


Subproblem 2
t + -4.5 = -6.652067348

Simplifying
t + -4.5 = -6.652067348

Reorder the terms:
-4.5 + t = -6.652067348

Solving
-4.5 + t = -6.652067348

Solving for variable 't'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + t = -6.652067348 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + t = -6.652067348 + 4.5
t = -6.652067348 + 4.5

Combine like terms: -6.652067348 + 4.5 = -2.152067348
t = -2.152067348

Simplifying
t = -2.152067348


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {11.152067348, -2.152067348}
Solution
t = {11.152067348, -2.152067348}

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