19.6=-4.9t^2+24.5t

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Solution for 19.6=-4.9t^2+24.5t equation: 


Simplifying
19.6 = -4.9t2 + 24.5t

Reorder the terms:
19.6 = 24.5t + -4.9t2

Solving
19.6 = 24.5t + -4.9t2

Solving for variable 't'.

Reorder the terms:
19.6 + -24.5t + 4.9t2 = 24.5t + -24.5t + -4.9t2 + 4.9t2

Combine like terms: 24.5t + -24.5t = 0.0
19.6 + -24.5t + 4.9t2 = 0.0 + -4.9t2 + 4.9t2
19.6 + -24.5t + 4.9t2 = -4.9t2 + 4.9t2

Combine like terms: -4.9t2 + 4.9t2 = 0.0
19.6 + -24.5t + 4.9t2 = 0.0

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

Divide each side by '4.9'.
4 + -5t + t2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -5t + -4 + t2 = 0 + -4

Reorder the terms:
4 + -4 + -5t + t2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -5t + t2 = 0 + -4
-5t + t2 = 0 + -4

Combine like terms: 0 + -4 = -4
-5t + t2 = -4

The t term is -5t.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5t + 6.25 + t2 = -4 + 6.25

Reorder the terms:
6.25 + -5t + t2 = -4 + 6.25

Combine like terms: -4 + 6.25 = 2.25
6.25 + -5t + t2 = 2.25

Factor a perfect square on the left side:
(t + -2.5)(t + -2.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(t + -2.5) equal to 1.5 and -1.5.


Subproblem 1
t + -2.5 = 1.5

Simplifying
t + -2.5 = 1.5

Reorder the terms:
-2.5 + t = 1.5

Solving
-2.5 + t = 1.5

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = 1.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = 1.5 + 2.5
t = 1.5 + 2.5

Combine like terms: 1.5 + 2.5 = 4
t = 4

Simplifying
t = 4


Subproblem 2
t + -2.5 = -1.5

Simplifying
t + -2.5 = -1.5

Reorder the terms:
-2.5 + t = -1.5

Solving
-2.5 + t = -1.5

Solving for variable 't'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + t = -1.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + t = -1.5 + 2.5
t = -1.5 + 2.5

Combine like terms: -1.5 + 2.5 = 1
t = 1

Simplifying
t = 1


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {4, 1}

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