15t(20t+5)=180

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Solution for 15t(20t+5)=180 equation: 


Simplifying
15t(20t + 5) = 180

Reorder the terms:
15t(5 + 20t) = 180
(5 * 15t + 20t * 15t) = 180
(75t + 300t2) = 180

Solving
75t + 300t2 = 180

Solving for variable 't'.

Reorder the terms:
-180 + 75t + 300t2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 75t + 300t2 = 0

Factor out the Greatest Common Factor (GCF), '15'.
15(-12 + 5t + 20t2) = 0

Ignore the factor 15.

Subproblem 1
Set the factor '(-12 + 5t + 20t2)' equal to zero and attempt to solve:

Simplifying
-12 + 5t + 20t2 = 0

Solving
-12 + 5t + 20t2 = 0

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

Divide each side by '20'.
-0.6 + 0.25t + t2 = 0.0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + 0.25t + 0.6 + t2 = 0.0 + 0.6

Reorder the terms:
-0.6 + 0.6 + 0.25t + t2 = 0.0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + 0.25t + t2 = 0.0 + 0.6
0.25t + t2 = 0.0 + 0.6

Combine like terms: 0.0 + 0.6 = 0.6
0.25t + t2 = 0.6

The t term is 0.25t.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25t + 0.015625 + t2 = 0.6 + 0.015625

Reorder the terms:
0.015625 + 0.25t + t2 = 0.6 + 0.015625

Combine like terms: 0.6 + 0.015625 = 0.615625
0.015625 + 0.25t + t2 = 0.615625

Factor a perfect square on the left side:
(t + 0.125)(t + 0.125) = 0.615625

Calculate the square root of the right side: 0.784617741

Break this problem into two subproblems by setting 
(t + 0.125) equal to 0.784617741 and -0.784617741.


Subproblem 1
t + 0.125 = 0.784617741

Simplifying
t + 0.125 = 0.784617741

Reorder the terms:
0.125 + t = 0.784617741

Solving
0.125 + t = 0.784617741

Solving for variable 't'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + t = 0.784617741 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + t = 0.784617741 + -0.125
t = 0.784617741 + -0.125

Combine like terms: 0.784617741 + -0.125 = 0.659617741
t = 0.659617741

Simplifying
t = 0.659617741


Subproblem 2
t + 0.125 = -0.784617741

Simplifying
t + 0.125 = -0.784617741

Reorder the terms:
0.125 + t = -0.784617741

Solving
0.125 + t = -0.784617741

Solving for variable 't'.

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

Add '-0.125' to each side of the equation.
0.125 + -0.125 + t = -0.784617741 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + t = -0.784617741 + -0.125
t = -0.784617741 + -0.125

Combine like terms: -0.784617741 + -0.125 = -0.909617741
t = -0.909617741

Simplifying
t = -0.909617741


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.659617741, -0.909617741}
Solution
t = {0.659617741, -0.909617741}

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