0=t^2-120t-180

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Solution for 0=t^2-120t-180 equation:


Simplifying
0 = t2 + -120t + -180

Reorder the terms:
0 = -180 + -120t + t2

Solving
0 = -180 + -120t + t2

Solving for variable 't'.

Combine like terms: 0 + 180 = 180
180 + 120t + -1t2 = -180 + -120t + t2 + 180 + 120t + -1t2

Reorder the terms:
180 + 120t + -1t2 = -180 + 180 + -120t + 120t + t2 + -1t2

Combine like terms: -180 + 180 = 0
180 + 120t + -1t2 = 0 + -120t + 120t + t2 + -1t2
180 + 120t + -1t2 = -120t + 120t + t2 + -1t2

Combine like terms: -120t + 120t = 0
180 + 120t + -1t2 = 0 + t2 + -1t2
180 + 120t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
180 + 120t + -1t2 = 0

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

Divide each side by '-1'.
-180 + -120t + t2 = 0

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + -120t + 180 + t2 = 0 + 180

Reorder the terms:
-180 + 180 + -120t + t2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + -120t + t2 = 0 + 180
-120t + t2 = 0 + 180

Combine like terms: 0 + 180 = 180
-120t + t2 = 180

The t term is -120t.  Take half its coefficient (-60).
Square it (3600) and add it to both sides.

Add '3600' to each side of the equation.
-120t + 3600 + t2 = 180 + 3600

Reorder the terms:
3600 + -120t + t2 = 180 + 3600

Combine like terms: 180 + 3600 = 3780
3600 + -120t + t2 = 3780

Factor a perfect square on the left side:
(t + -60)(t + -60) = 3780

Calculate the square root of the right side: 61.481704596

Break this problem into two subproblems by setting 
(t + -60) equal to 61.481704596 and -61.481704596.

Subproblem 1

t + -60 = 61.481704596 Simplifying t + -60 = 61.481704596 Reorder the terms: -60 + t = 61.481704596 Solving -60 + t = 61.481704596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + t = 61.481704596 + 60 Combine like terms: -60 + 60 = 0 0 + t = 61.481704596 + 60 t = 61.481704596 + 60 Combine like terms: 61.481704596 + 60 = 121.481704596 t = 121.481704596 Simplifying t = 121.481704596

Subproblem 2

t + -60 = -61.481704596 Simplifying t + -60 = -61.481704596 Reorder the terms: -60 + t = -61.481704596 Solving -60 + t = -61.481704596 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + t = -61.481704596 + 60 Combine like terms: -60 + 60 = 0 0 + t = -61.481704596 + 60 t = -61.481704596 + 60 Combine like terms: -61.481704596 + 60 = -1.481704596 t = -1.481704596 Simplifying t = -1.481704596

Solution

The solution to the problem is based on the solutions from the subproblems. t = {121.481704596, -1.481704596}

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