Question

The equation -16^2 + 144 gives the height, in feet, of a toy rocket t seconds after it was launched up into the air. How long will it take for the rocket to return to the ground? Solve the quadratic equatio-16^2 + 144 = 0 by factoring to solve the problem.

Answer 1

In this problem, we want to apply the use of a quadratic function.

We are given the following equation to model the path of a rocket:

`-16t^2+144t=0`

To find how long it takes the rocket to hit the ground, the equation must be equal to 0. Luckily, that's already true.

Begin by identifying the greatest common factor of

`-16t^2\text{ and }144t.`

We see that both terms have 16 and t in common:

`GCF(-16t^2,144t)=16t`

We can factor that expression out from each term to get:

`-16t^2+144t=16t(-t+9)`

Now we have the equation:

`16t(-t+9)=0`

Applying the zero product property, we can split this into two equations:

`16t=0\text{ and }-t+9=0`

Solve the first equation by dividing by 16 on both sides:

`\begin{gathered} 16t=0 \\ \\ t=0 \end{gathered}`

Solve the second equation by adding t to both sides:

`\begin{gathered} -t+9=0 \\ \\ 9=t \end{gathered}`

Since we want to know how long it takes the rocket to hit the ground after takeoff, we can ignore the first solution, which represents 0 seconds.

It took the rocket 9 seconds to hit the ground.