# Video: Solving Quadratic Equations Involving Projectiles

The height of a missile above its launch point can be found using 𝑠 = 𝑢𝑡 − 4.9𝑡², where 𝑠 m is the height above the launch point, 𝑢 m/s is the vertical launch speed, and 𝑡 s is the time after launch. A missile is launched vertically upwards with a speed of 49 m/s. At what times will it be 44.1 m above its launch point?

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### Video Transcript

The height of a missile above its launch point can be found using 𝑠 equals 𝑢𝑡 minus 4.9 𝑡 squared, where 𝑠 metres is the height above the launch point, 𝑢 metres per second is the vertical launch speed, and 𝑡 seconds is the time after launch. A missile is launched vertically upwards with a speed of 49 metres per second. At what times will it be 44.1 metres above its launch point?

So here we have our equation and it’s asking us to find times, at which it will be 44.1 metres above its launch point. So we are solving for 𝑡. So we want 𝑡 to be the only variable left in the equation so we can solve for it. So we need to plug in numbers for 𝑠 and for 𝑢. It’s said that the missile was launched at a speed of 49 metres per second and 𝑢 is the vertical launch speed. So Let’s plug in 49 for 𝑢. It also tells us it will be 44.1 metres above the launch point and that is 𝑠, so we can replace 𝑠 with 44.1.

So in order to solve for 𝑡, we’re going to have to factor. And when we factor, we want the leaving term to be positive. So when we put a trinomial — a polynomial — in descending order, the first term needs to be positive. So that 4.9 𝑡 squared should be our first term because it has the highest exponent. And here it’s negative, so let’s bring everything over to the left-hand side. So we need to add 4.9 𝑡 squared to both sides as well as subtracting 49 𝑡 to both sides. So we are left with 4.9 𝑡 squared minus 49 𝑡 plus 44.1 equals zero.

So the first thing we should do is check for greatest common factor, and we have 4.9 times 𝑡 squared minus 10 𝑡 plus nine equals zero. So in order to factor, we have to figure out what two numbers multiply to be positive nine and add to be negative 10. Numbers that multiply will be nine or three and three and nine and one. Now they add to be a negative number, but multiply to be a positive number, which means both numbers need to be negative. So which is said adds to be negative 10? That would be negative nine and negative one.

So we have 4.9 times 𝑡 minus nine times 𝑡 minus one equals zero. And we set each factor equal to zero and solve. Setting 4.9 equal to zero doesn’t actually help us because there is no variable to solve for. So now solving 𝑡 minus nine equals zero, we need to add nine to both sides and we get 𝑡 equals nine. And now for the next one, we need to add one to both sides. So we get 𝑡 equals one. So this means that the missile is at a height of 44.1 metres at nine seconds and one second.