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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 8: Topic A: Lessons 8-10: Parabolas intro

# Quadratic word problems (factored form)

Solving a problem where a quadratic function (given in factored form) models the height of a launched rocket.

## Want to join the conversation?

• I can solve these equations but I don't understand what each expression means.
x+2=0, x=-2.
x-18 =0, x=18.
(-2+18)/2 =8.
-4(8+2)(8-18)=400

What does -4 mean in this circumstance? and what are (x+2) (x-18) mean?
• If you start from - 4(x+2)(x-18) and expand it to f(x) = - 4(x^2 - 16x - 36) or f(x) = -4x^2 + 64x + 144, the -4 has to do with the force of gravity trying to pull the rocket back to the ground, the 64 has to do with the initial velocity that the rocket was launched at, and the 144 has to do with the initial height of the rocket. The equation often uses t instead of x because t would stand for time and f(t) is height above ground. The -2 and the 18 are the solutions to the quadratic function, which in this case means that this will be either a real (18) or hypothetical (-2) time when the rocket is on ground level. the -2 is hypothetical because it is behind where the rocket is launched, so it could be shown as a dashed line from time -2 to 0 which fits the function, but not the situation.
• At isn't it supposed to be 20 seconds as the launch starts at x=-2 and goes to 18 which makes the total time in the air 20 seconds and make the rocket land after 20 seconds?
• It would be 20 seconds if the rocket was launched from the ground. In this case, the rocket was launched from a raised platform. The x=-2 is basically denoting that 2 seconds of flight were saved by launching from the platform rather than the ground.
Hope this helps.
• so if the equation is h(x) = -4(x - 18), how would you find the first x value?
• if y=0 then either -4 or (x-18) equals zero. -4 cannot equal zero, so (x-18) has to. So x=18 and there is no second x.
• I've come to notice that when you are trying to find the y coordinate for the vertex that the expressions within the parentheses are evaluated as a number and the negative of the number. For example in this problem the evaluation of the parentheses was 10 and -10. Is it coincidental and if not what is the reasoning behind this remarkable observation?
• It is what should be expected because you found the midpoint between -2 and 18. The distance between these two is 20, so the midpoint was at 8 which should be 10 units away from the two points. Thus, one of the points mush be + 10 away, and the other must be - 10 away. Even if you change points such as 6 and 18, you would find the midpoint as (6 + 18)/2 = 12. Since 18-6 = 12, they are twelve units away from each other, so dividing by two gives + 6 and -6 from the midpoint. There is no coincidence about it.
• Umm... Guys, what if we were asked to give the model?
• Like graphing it? It'll look something like a curve with the 0's as -2 and 18.
• At , In the question it asked "how many seconds after being launched will the rocket reach its maximum height?" So what i thought of it was that, since the total air time of the rocket was 18 seconds so it would mean that at 9 seconds it will be at its maximum point so shouldn't 9 seconds be the correct answer?
• Ok. I will solve it my way, and hope you can get something from it.
1)What is the height of the rocket at the time of launch ? Well, the time at launch is x=0, so h(0)=4(0+2)(0-18) , which equals -4(2)(-18)=(144), so the height at the time of launch is 144 assuming x=0 is the ground.

2)How many seconds after the launch will the rocket hit the ground ? The rocket will reach the ground when h(x)=0, which will hapen in two points. we can apply Bhaskara´s formula in order to find zeroes or just take a look at the function in this case.
h(x) = -4(x+2)(x-18), so h(x)=0 when (x+2)=0 or when (x-18)=0 .So h(x)=0 when x=-2 or when x=18. Lets check for extraneous solutions by plugging the results back in the function. -2 is not a valid solution, because the rocket cannot fall to the ground before it has taken of, so that leaves us with x=18 as the only valid answer.I checked all my answers with a graphic calculator.
• My thinking is that the midpoint X should be 10. It's 20 units between them where X=seconds. If you follow the graph, that would mean that the rocket after 0 seconds launch would just appear so many meters above the ground. The rocket's path must start at 0 meters and 0 seconds. It would proceed with the same curve, and maximum height, but it cant launch 2 seconds in the past. Unless you actually think of it like this: two seconds after launch it reached Y meters. Am I making any sense?
• While I like your thinking process, you make several incorrect assumptions. In the problem, it clearly states that it is launched from a platform, so your assumption that it must start at 0 meters and 0 seconds is just incorrect. If it were a function to be graphed independent of the word problem, both sides of the parabola would extend to negative infinity. However, within the context of the word problem, the domain would be from 0 to 18 seconds when it hit the ground. So the word problem would cut off all of the parabola less than 0 (including the part that is reflected from 16-18 seconds at -2-0) as well as the part greater than 18. Being launched from a platform that is 144 m about the ground gives a starting point at (0,144) not (0,0), and the symmetrical point would be at (16,144) which again gives the same midpoint at 8. The part between 16 and 18 just does not have a part that is reflected across the line of symmetry. While the -2 point is not within the domain of the word problem, it is still within the graph of the function and is the imaginary reflection of (18.0) which still gives midpoint at 8. It not that the midpoint is 10 units from the y intercept, it is 10 units from the endpoint 18 and 18-10 = 8. So what you really are attempting to do is shift the whole thing 2 units to the left so that you will begin at (0,0) because you do not like the fact that it was launched from a platform.
• Ive watched it many times. but i still dont understand how to solve for x.
• That is a great question! The two Xs in the parentheses are going to be opposite of what they are being added/subtracted to since the first X in the problem is 0. If you are asking to answer how many seconds after the rocket was launched it hit the ground then here you go as well. Since after the rocket we are going forward in time, between the two answers of x=-2 and x=+18 the only one reasonable enough would be x=18 then that would be the answer. Not sure if I confused you more or not but that is all I have, I also had looked back at the video, amazing question.
• The rocket launch in -2 sec, and it falls in 18 sec, I think the answer should be 20 and it's not 18.

Is my thinking correct?