### Video Transcript

A shell was fired from a mortar along a trajectory described by the equation ๐ฆ equals 0.19 plus 0.31๐ฅ minus 0.5๐ฅ squared, where ๐ฆ is the height of the shell above the ground in kilometres when it has travelled a horizontal distance of ๐ฅ kilometres. Find the horizontal distance covered by the shell before it hit the ground.

So if we would fire a shell, it would go up in the air and then eventually hit the ground. And ๐ฆ represents the height of that shell, so when it hits the ground, ๐ฆ would be zero and ๐ฅ represents how far it would travel horizontally. So if we take our equation and we put it in descending order, we would put the ๐ฅ squared term first, then the ๐ฅ, and then the constant. So from here, we will know the distance travelled before it hit the ground and it will hit the ground when ๐ฆ is zero. So we said that itโs equal to zero.

So we can solve this quadratic equation a few ways, but the best way since thereโs lots of decimals would be the quadratic formula. The quadratic formula is negative ๐ plus or minus the square root of ๐ squared minus four ๐๐ all over two ๐. ๐ is the coefficient in front of ๐ฅ squared, ๐ is the coefficient in front of ๐ฅ, and ๐ is your constant. So taking our values, we can solve for ๐ฅ, the distance that it travelled horizontally, by plugging these into that quadratic formula. Now that theyโve been plugged in, letโs evaluate.

So after squaring our decimal and multiplying four ๐๐, letโs simplify this more. And now letโs take the square root and we get 0.31 plus or minus 0.69. But before we get too far, letโs take a few steps back and look at our signs. The negative 0.31 cancelled with the negative with the one, so thatโs why 0.31 is positive. And then since we have a plus and a minus, it wouldnโt matter if we would bring the negative up with it. It would still have a plus and a minus.

So we need to take 0.31 and add 0.69 and also subtract 0.69. So we get ๐ฅ equals one or ๐ฅ equals negative 0.38. However, this is a horizontal distance, so itโs not going to travel at negative amount. So that means after the shell was fired, it travelled a horizontal distance of one kilometre before it hit the ground.