0=t^2+90t-120

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Solution for 0=t^2+90t-120 equation:


Simplifying
0 = t2 + 90t + -120

Reorder the terms:
0 = -120 + 90t + t2

Solving
0 = -120 + 90t + t2

Solving for variable 't'.

Combine like terms: 0 + 120 = 120
120 + -90t + -1t2 = -120 + 90t + t2 + 120 + -90t + -1t2

Reorder the terms:
120 + -90t + -1t2 = -120 + 120 + 90t + -90t + t2 + -1t2

Combine like terms: -120 + 120 = 0
120 + -90t + -1t2 = 0 + 90t + -90t + t2 + -1t2
120 + -90t + -1t2 = 90t + -90t + t2 + -1t2

Combine like terms: 90t + -90t = 0
120 + -90t + -1t2 = 0 + t2 + -1t2
120 + -90t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
120 + -90t + -1t2 = 0

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

Divide each side by '-1'.
-120 + 90t + t2 = 0

Move the constant term to the right:

Add '120' to each side of the equation.
-120 + 90t + 120 + t2 = 0 + 120

Reorder the terms:
-120 + 120 + 90t + t2 = 0 + 120

Combine like terms: -120 + 120 = 0
0 + 90t + t2 = 0 + 120
90t + t2 = 0 + 120

Combine like terms: 0 + 120 = 120
90t + t2 = 120

The t term is 90t.  Take half its coefficient (45).
Square it (2025) and add it to both sides.

Add '2025' to each side of the equation.
90t + 2025 + t2 = 120 + 2025

Reorder the terms:
2025 + 90t + t2 = 120 + 2025

Combine like terms: 120 + 2025 = 2145
2025 + 90t + t2 = 2145

Factor a perfect square on the left side:
(t + 45)(t + 45) = 2145

Calculate the square root of the right side: 46.314144708

Break this problem into two subproblems by setting 
(t + 45) equal to 46.314144708 and -46.314144708.

Subproblem 1

t + 45 = 46.314144708 Simplifying t + 45 = 46.314144708 Reorder the terms: 45 + t = 46.314144708 Solving 45 + t = 46.314144708 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-45' to each side of the equation. 45 + -45 + t = 46.314144708 + -45 Combine like terms: 45 + -45 = 0 0 + t = 46.314144708 + -45 t = 46.314144708 + -45 Combine like terms: 46.314144708 + -45 = 1.314144708 t = 1.314144708 Simplifying t = 1.314144708

Subproblem 2

t + 45 = -46.314144708 Simplifying t + 45 = -46.314144708 Reorder the terms: 45 + t = -46.314144708 Solving 45 + t = -46.314144708 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-45' to each side of the equation. 45 + -45 + t = -46.314144708 + -45 Combine like terms: 45 + -45 = 0 0 + t = -46.314144708 + -45 t = -46.314144708 + -45 Combine like terms: -46.314144708 + -45 = -91.314144708 t = -91.314144708 Simplifying t = -91.314144708

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.314144708, -91.314144708}

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