54=-.8t^2+14.4t+2

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Solution for 54=-.8t^2+14.4t+2 equation:


Simplifying
54 = -0.8t2 + 14.4t + 2

Reorder the terms:
54 = 2 + 14.4t + -0.8t2

Solving
54 = 2 + 14.4t + -0.8t2

Solving for variable 't'.

Combine like terms: 54 + -2 = 52
52 + -14.4t + 0.8t2 = 2 + 14.4t + -0.8t2 + -2 + -14.4t + 0.8t2

Reorder the terms:
52 + -14.4t + 0.8t2 = 2 + -2 + 14.4t + -14.4t + -0.8t2 + 0.8t2

Combine like terms: 2 + -2 = 0
52 + -14.4t + 0.8t2 = 0 + 14.4t + -14.4t + -0.8t2 + 0.8t2
52 + -14.4t + 0.8t2 = 14.4t + -14.4t + -0.8t2 + 0.8t2

Combine like terms: 14.4t + -14.4t = 0.0
52 + -14.4t + 0.8t2 = 0.0 + -0.8t2 + 0.8t2
52 + -14.4t + 0.8t2 = -0.8t2 + 0.8t2

Combine like terms: -0.8t2 + 0.8t2 = 0.0
52 + -14.4t + 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'.
65 + -18t + t2 = 0

Move the constant term to the right:

Add '-65' to each side of the equation.
65 + -18t + -65 + t2 = 0 + -65

Reorder the terms:
65 + -65 + -18t + t2 = 0 + -65

Combine like terms: 65 + -65 = 0
0 + -18t + t2 = 0 + -65
-18t + t2 = 0 + -65

Combine like terms: 0 + -65 = -65
-18t + t2 = -65

The t term is -18t.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18t + 81 + t2 = -65 + 81

Reorder the terms:
81 + -18t + t2 = -65 + 81

Combine like terms: -65 + 81 = 16
81 + -18t + t2 = 16

Factor a perfect square on the left side:
(t + -9)(t + -9) = 16

Calculate the square root of the right side: 4

Break this problem into two subproblems by setting 
(t + -9) equal to 4 and -4.

Subproblem 1

t + -9 = 4 Simplifying t + -9 = 4 Reorder the terms: -9 + t = 4 Solving -9 + t = 4 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = 4 + 9 Combine like terms: -9 + 9 = 0 0 + t = 4 + 9 t = 4 + 9 Combine like terms: 4 + 9 = 13 t = 13 Simplifying t = 13

Subproblem 2

t + -9 = -4 Simplifying t + -9 = -4 Reorder the terms: -9 + t = -4 Solving -9 + t = -4 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + t = -4 + 9 Combine like terms: -9 + 9 = 0 0 + t = -4 + 9 t = -4 + 9 Combine like terms: -4 + 9 = 5 t = 5 Simplifying t = 5

Solution

The solution to the problem is based on the solutions from the subproblems. t = {13, 5}

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