0=-9t^2+72t+90

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Solution for 0=-9t^2+72t+90 equation: 


Simplifying
0 = -9t2 + 72t + 90

Reorder the terms:
0 = 90 + 72t + -9t2

Solving
0 = 90 + 72t + -9t2

Solving for variable 't'.

Combine like terms: 0 + -90 = -90
-90 + -72t + 9t2 = 90 + 72t + -9t2 + -90 + -72t + 9t2

Reorder the terms:
-90 + -72t + 9t2 = 90 + -90 + 72t + -72t + -9t2 + 9t2

Combine like terms: 90 + -90 = 0
-90 + -72t + 9t2 = 0 + 72t + -72t + -9t2 + 9t2
-90 + -72t + 9t2 = 72t + -72t + -9t2 + 9t2

Combine like terms: 72t + -72t = 0
-90 + -72t + 9t2 = 0 + -9t2 + 9t2
-90 + -72t + 9t2 = -9t2 + 9t2

Combine like terms: -9t2 + 9t2 = 0
-90 + -72t + 9t2 = 0

Factor out the Greatest Common Factor (GCF), '9'.
9(-10 + -8t + t2) = 0

Ignore the factor 9.

Subproblem 1
Set the factor '(-10 + -8t + t2)' equal to zero and attempt to solve:

Simplifying
-10 + -8t + t2 = 0

Solving
-10 + -8t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -8t + 10 + t2 = 0 + 10

Reorder the terms:
-10 + 10 + -8t + t2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -8t + t2 = 0 + 10
-8t + t2 = 0 + 10

Combine like terms: 0 + 10 = 10
-8t + t2 = 10

The t term is -8t.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8t + 16 + t2 = 10 + 16

Reorder the terms:
16 + -8t + t2 = 10 + 16

Combine like terms: 10 + 16 = 26
16 + -8t + t2 = 26

Factor a perfect square on the left side:
(t + -4)(t + -4) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(t + -4) equal to 5.099019514 and -5.099019514.


Subproblem 1
t + -4 = 5.099019514

Simplifying
t + -4 = 5.099019514

Reorder the terms:
-4 + t = 5.099019514

Solving
-4 + t = 5.099019514

Solving for variable 't'.

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

Add '4' to each side of the equation.
-4 + 4 + t = 5.099019514 + 4

Combine like terms: -4 + 4 = 0
0 + t = 5.099019514 + 4
t = 5.099019514 + 4

Combine like terms: 5.099019514 + 4 = 9.099019514
t = 9.099019514

Simplifying
t = 9.099019514


Subproblem 2
t + -4 = -5.099019514

Simplifying
t + -4 = -5.099019514

Reorder the terms:
-4 + t = -5.099019514

Solving
-4 + t = -5.099019514

Solving for variable 't'.

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

Add '4' to each side of the equation.
-4 + 4 + t = -5.099019514 + 4

Combine like terms: -4 + 4 = 0
0 + t = -5.099019514 + 4
t = -5.099019514 + 4

Combine like terms: -5.099019514 + 4 = -1.099019514
t = -1.099019514

Simplifying
t = -1.099019514


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {9.099019514, -1.099019514}
Solution
t = {9.099019514, -1.099019514}

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