Video: Creating and Using Quadratic Equations in More Than One Variable

A sprinkler is used to water a garden. The height ℎ, in meters, of the jet of water a horizontal distance 𝑥 meters from the sprinkler can be found using the quadratic equation ℎ = 𝑥 − (𝑥²/4). Write an equation that will give the distance, in meters, between the sprinkler and the point where the jet hits the ground. How far from the sprinkler does the jet hit the ground?

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Video Transcript

A sprinkler is used to water a garden. The height ℎ, in meters, of the jet of water a horizontal distance 𝑥 meters from the sprinkler can be found using the quadratic equation ℎ equals 𝑥 minus 𝑥 squared over four. Write an equation that will give the distance, in meters, between the sprinkler and the point where the jet hits the ground. How far from the sprinkler does the jet hit the ground?

So for the first part, what we need to do is actually write an equation that will give us the distance in meters between the sprinkler and the point where the jet hits the ground. Well, we’re told in the question that the height is ℎ. Well, if the height is ℎ and we want to find out when actually the jet hits the ground, then at this point, we’re gonna know that ℎ is gonna be equal to zero. And we know that because if it’s hid in the ground, then clearly the height from the ground is going to be equal to zero.

Okay, so now, let’s substitute this into the quadratic equation that we had above. Okay, so when we’ve done that, what we actually have done is written an equation that will give the distance in meters between the sprinkler and the point where the jet hits the ground. This equation is 𝑥 minus 𝑥 squared over four is equal to zero.

Okay, fab, we’ve solved the first part of the question. Now, let’s move on to the second part. So now, for the second part of the question, we need to find out how far from the sprinkler does the jet hit the ground. So what we’re actually trying to do is solve for 𝑥. And that’s because the horizontal distance is represented by 𝑥, so our horizontal distance from the sprinkler.

Okay, so now, let’s solve our equation. So we’ve got 𝑥 minus 𝑥 squared over four equals zero cause we got that from the first part of the question. So the first thing we’re gonna do is actually multiply through our equation by four because obviously we’ve got a fraction — we got 𝑥 squared over four — and we actually want to remove that.

So to do that, we can multiply both sides of our equation by four. And when we do that, we get four 𝑥 because four multiplied by 𝑥 is four 𝑥 minus 𝑥 squared. And that’s cause 𝑥 squared over four multiplied by four just gives us 𝑥 squared is equal to zero again cause zero multiplied by four is zero.

Okay, so now, what we’re gonna do? Well, now, we’re actually going to factor the small thing that we have on the left-hand side, so our expression which is four 𝑥 minus 𝑥 squared. And when we actually factor this, what we’re gonna get is 𝑥 outside the parentheses. And that’s because 𝑥 is in both four 𝑥 and negative 𝑥 squared. And then, inside the parentheses, we have four minus 𝑥. And that’s because 𝑥 multiplied by four gives us our four 𝑥 and 𝑥 multiplied by negative 𝑥 gives us negative 𝑥 squared.

So therefore, 𝑥 is gonna be equal to zero or four. And we got those two values for 𝑥 because if 𝑥 was equal to zero, so that’ll be zero multiplied by four minus zero. Well, anything multiplied by zero is just zero. And if 𝑥 was equal to four, we’d have four multiplied by then four minus four. Well, four minus four is just zero. So it’ll be four multiplied by zero. Well, again, this would equal zero.

Okay, so that’s how we found our 𝑥-values. But we’ve got two 𝑥-values: zero and four. Which one is going to be the correct one? Okay, so we know that it’s gonna be 𝑥 equals four, not 𝑥 equals zero because we can’t have 𝑥 equals zero because if we did, that would be where the sprinkler is. And we actually want to see how far the jet travels to hit the ground again.

Therefore, we can say that the distance travelled from the sprinkler of the jet before it hits the ground is going to be equal to four meters and that’s because 𝑥 was equal to four.

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