0=24t-.8t^2-90

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Solution for 0=24t-.8t^2-90 equation:


Simplifying
0 = 24t + -0.8t2 + -90

Reorder the terms:
0 = -90 + 24t + -0.8t2

Solving
0 = -90 + 24t + -0.8t2

Solving for variable 't'.

Combine like terms: 0 + 90 = 90
90 + -24t + 0.8t2 = -90 + 24t + -0.8t2 + 90 + -24t + 0.8t2

Reorder the terms:
90 + -24t + 0.8t2 = -90 + 90 + 24t + -24t + -0.8t2 + 0.8t2

Combine like terms: -90 + 90 = 0
90 + -24t + 0.8t2 = 0 + 24t + -24t + -0.8t2 + 0.8t2
90 + -24t + 0.8t2 = 24t + -24t + -0.8t2 + 0.8t2

Combine like terms: 24t + -24t = 0
90 + -24t + 0.8t2 = 0 + -0.8t2 + 0.8t2
90 + -24t + 0.8t2 = -0.8t2 + 0.8t2

Combine like terms: -0.8t2 + 0.8t2 = 0.0
90 + -24t + 0.8t2 = 0.0

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

Divide each side by '0.8'.
112.5 + -30t + t2 = 0

Move the constant term to the right:

Add '-112.5' to each side of the equation.
112.5 + -30t + -112.5 + t2 = 0 + -112.5

Reorder the terms:
112.5 + -112.5 + -30t + t2 = 0 + -112.5

Combine like terms: 112.5 + -112.5 = 0.0
0.0 + -30t + t2 = 0 + -112.5
-30t + t2 = 0 + -112.5

Combine like terms: 0 + -112.5 = -112.5
-30t + t2 = -112.5

The t term is -30t.  Take half its coefficient (-15).
Square it (225) and add it to both sides.

Add '225' to each side of the equation.
-30t + 225 + t2 = -112.5 + 225

Reorder the terms:
225 + -30t + t2 = -112.5 + 225

Combine like terms: -112.5 + 225 = 112.5
225 + -30t + t2 = 112.5

Factor a perfect square on the left side:
(t + -15)(t + -15) = 112.5

Calculate the square root of the right side: 10.606601718

Break this problem into two subproblems by setting 
(t + -15) equal to 10.606601718 and -10.606601718.

Subproblem 1

t + -15 = 10.606601718 Simplifying t + -15 = 10.606601718 Reorder the terms: -15 + t = 10.606601718 Solving -15 + t = 10.606601718 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + t = 10.606601718 + 15 Combine like terms: -15 + 15 = 0 0 + t = 10.606601718 + 15 t = 10.606601718 + 15 Combine like terms: 10.606601718 + 15 = 25.606601718 t = 25.606601718 Simplifying t = 25.606601718

Subproblem 2

t + -15 = -10.606601718 Simplifying t + -15 = -10.606601718 Reorder the terms: -15 + t = -10.606601718 Solving -15 + t = -10.606601718 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + t = -10.606601718 + 15 Combine like terms: -15 + 15 = 0 0 + t = -10.606601718 + 15 t = -10.606601718 + 15 Combine like terms: -10.606601718 + 15 = 4.393398282 t = 4.393398282 Simplifying t = 4.393398282

Solution

The solution to the problem is based on the solutions from the subproblems. t = {25.606601718, 4.393398282}

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