45(t)=-16t^2+5t+105

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Solution for 45(t)=-16t^2+5t+105 equation:


Simplifying
45(t) = -16t2 + 5t + 105

Reorder the terms:
45t = 105 + 5t + -16t2

Solving
45t = 105 + 5t + -16t2

Solving for variable 't'.

Reorder the terms:
-105 + 45t + -5t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2

Combine like terms: 45t + -5t = 40t
-105 + 40t + 16t2 = 105 + 5t + -16t2 + -105 + -5t + 16t2

Reorder the terms:
-105 + 40t + 16t2 = 105 + -105 + 5t + -5t + -16t2 + 16t2

Combine like terms: 105 + -105 = 0
-105 + 40t + 16t2 = 0 + 5t + -5t + -16t2 + 16t2
-105 + 40t + 16t2 = 5t + -5t + -16t2 + 16t2

Combine like terms: 5t + -5t = 0
-105 + 40t + 16t2 = 0 + -16t2 + 16t2
-105 + 40t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-105 + 40t + 16t2 = 0

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

Divide each side by '16'.
-6.5625 + 2.5t + t2 = 0

Move the constant term to the right:

Add '6.5625' to each side of the equation.
-6.5625 + 2.5t + 6.5625 + t2 = 0 + 6.5625

Reorder the terms:
-6.5625 + 6.5625 + 2.5t + t2 = 0 + 6.5625

Combine like terms: -6.5625 + 6.5625 = 0.0000
0.0000 + 2.5t + t2 = 0 + 6.5625
2.5t + t2 = 0 + 6.5625

Combine like terms: 0 + 6.5625 = 6.5625
2.5t + t2 = 6.5625

The t term is 2.5t.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5t + 1.5625 + t2 = 6.5625 + 1.5625

Reorder the terms:
1.5625 + 2.5t + t2 = 6.5625 + 1.5625

Combine like terms: 6.5625 + 1.5625 = 8.125
1.5625 + 2.5t + t2 = 8.125

Factor a perfect square on the left side:
(t + 1.25)(t + 1.25) = 8.125

Calculate the square root of the right side: 2.850438563

Break this problem into two subproblems by setting 
(t + 1.25) equal to 2.850438563 and -2.850438563.

Subproblem 1

t + 1.25 = 2.850438563 Simplifying t + 1.25 = 2.850438563 Reorder the terms: 1.25 + t = 2.850438563 Solving 1.25 + t = 2.850438563 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = 2.850438563 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 2.850438563 + -1.25 t = 2.850438563 + -1.25 Combine like terms: 2.850438563 + -1.25 = 1.600438563 t = 1.600438563 Simplifying t = 1.600438563

Subproblem 2

t + 1.25 = -2.850438563 Simplifying t + 1.25 = -2.850438563 Reorder the terms: 1.25 + t = -2.850438563 Solving 1.25 + t = -2.850438563 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + t = -2.850438563 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -2.850438563 + -1.25 t = -2.850438563 + -1.25 Combine like terms: -2.850438563 + -1.25 = -4.100438563 t = -4.100438563 Simplifying t = -4.100438563

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.600438563, -4.100438563}

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