45t^2-72t+24=0

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Solution for 45t^2-72t+24=0 equation: 


Simplifying
45t2 + -72t + 24 = 0

Reorder the terms:
24 + -72t + 45t2 = 0

Solving
24 + -72t + 45t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '3'.
3(8 + -24t + 15t2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(8 + -24t + 15t2)' equal to zero and attempt to solve:

Simplifying
8 + -24t + 15t2 = 0

Solving
8 + -24t + 15t2 = 0

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

Divide each side by '15'.
0.5333333333 + -1.6t + t2 = 0

Move the constant term to the right:

Add '-0.5333333333' to each side of the equation.
0.5333333333 + -1.6t + -0.5333333333 + t2 = 0 + -0.5333333333

Reorder the terms:
0.5333333333 + -0.5333333333 + -1.6t + t2 = 0 + -0.5333333333

Combine like terms: 0.5333333333 + -0.5333333333 = 0.0000000000
0.0000000000 + -1.6t + t2 = 0 + -0.5333333333
-1.6t + t2 = 0 + -0.5333333333

Combine like terms: 0 + -0.5333333333 = -0.5333333333
-1.6t + t2 = -0.5333333333

The t term is -1.6t.  Take half its coefficient (-0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
-1.6t + 0.64 + t2 = -0.5333333333 + 0.64

Reorder the terms:
0.64 + -1.6t + t2 = -0.5333333333 + 0.64

Combine like terms: -0.5333333333 + 0.64 = 0.1066666667
0.64 + -1.6t + t2 = 0.1066666667

Factor a perfect square on the left side:
(t + -0.8)(t + -0.8) = 0.1066666667

Calculate the square root of the right side: 0.326598632

Break this problem into two subproblems by setting 
(t + -0.8) equal to 0.326598632 and -0.326598632.


Subproblem 1
t + -0.8 = 0.326598632

Simplifying
t + -0.8 = 0.326598632

Reorder the terms:
-0.8 + t = 0.326598632

Solving
-0.8 + t = 0.326598632

Solving for variable 't'.

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

Add '0.8' to each side of the equation.
-0.8 + 0.8 + t = 0.326598632 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + t = 0.326598632 + 0.8
t = 0.326598632 + 0.8

Combine like terms: 0.326598632 + 0.8 = 1.126598632
t = 1.126598632

Simplifying
t = 1.126598632


Subproblem 2
t + -0.8 = -0.326598632

Simplifying
t + -0.8 = -0.326598632

Reorder the terms:
-0.8 + t = -0.326598632

Solving
-0.8 + t = -0.326598632

Solving for variable 't'.

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

Add '0.8' to each side of the equation.
-0.8 + 0.8 + t = -0.326598632 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + t = -0.326598632 + 0.8
t = -0.326598632 + 0.8

Combine like terms: -0.326598632 + 0.8 = 0.473401368
t = 0.473401368

Simplifying
t = 0.473401368


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.126598632, 0.473401368}
Solution
t = {1.126598632, 0.473401368}

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