Tutor.com Quadratic Equations Session

Apr. 30, 2012

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Session Transcript - Math - Algebra, 4/5/2012 12:45AM - Tutor.comSession Date: 4/5/2012 12:45AM
Length: 106.2 minute(s)
Subject: Math - Algebra


System Message
[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Guest (Customer)
[00:00:08] Consider the parabolic paths of two soccer penalty kicks, represented in the graph at right. Kick #1 covers a horizontal distance of 20 yds and reaches a maximum height of 9yds. Kick #2 covers a horizontal distance of 24yds, but only reaches a maximum height of 6yds. Find an equation for the path of each kick? How far off the ground would they be when they collide? [ File > http://0-lhh.tutor.com.catalog.multcolib.org/SharedSessionFiles/501955c1-27d9-4ea3-8013-6947cc99f605_tn.jpg ]

Blythe L (Tutor)
[00:00:21] Welcome to Tutor.com ! Let's get started.
[00:00:23] hi

Guest (Customer)
[00:00:28] hello

Blythe L (Tutor)
[00:03:23] okay, here are the two parabolas
[00:03:33] does it make sense how I drew them so far?

Guest (Customer)
[00:03:44] by the info given

Blythe L (Tutor)
[00:04:09] so, now, we need to make the two equations for the parabolas
[00:04:34] and then we set them equal to each other and then solve
[00:05:08] so, let's use the vertex and another point to find the equation
[00:05:13] do you know how to do that?

Guest (Customer)
[00:05:24] no

Blythe L (Tutor)
[00:05:37] okay, I'll show you ...one sec...
[00:06:51] this is the vertex form of the equation
[00:07:44] the vertex is (h, k)
[00:08:22] so, I plug in (12, 6) for the vertex

Guest (Customer)
[00:08:33] ok

Blythe L (Tutor)
[00:08:49] and also, since (0,0) is another point on the parabola, then we can plug in that point for x,y and then we can find the "a"
[00:10:43] so, the "a" is -1/24
[00:11:04] and so the equation of the first one is y=-1/24 ( x-12)^2 +6
[00:12:03] okay, so that's the first one
[00:12:12] now, do you want to try to find the second equation

Guest (Customer)
[00:12:12] ok thanks

Blythe L (Tutor)
[00:12:26] using the (10,9) and (0,0) ?
[00:12:36] I'll tell you if there's a mistake
[00:13:31] wait, that's not how you do it
[00:13:52] do you see where the "h" and "k" is in the equation?
[00:14:03] the "h" is the x value of the vertex, 10
[00:14:12] the "k" is the y value of the vertex, 9
[00:14:35] and you will plug in (0,0) as the x,y in the equation
[00:17:23] so, that is the equation for the second one
[00:17:26] does it make sense?

Guest (Customer)
[00:17:34] kind of

Blythe L (Tutor)
[00:18:27] I plugged in (0,0) for (x,y) and I plugged in (10,9) for (h,k)....that was all just to find the "a" (the coefficient in front of the x)
[00:18:46] now, to find the point of intersection...
[00:18:58] you will need to set the equations equal to each other
[00:19:03] and solve for "x"
[00:19:16] do you understand what I'm talking about?

Guest (Customer)
[00:19:29] sort of

Blythe L (Tutor)
[00:20:45] set the two equations equal to each other and you are going to solve for "x"
[00:22:15] okay, now I have written them for you on the second whiteboard
[00:22:30] you will need to use this equation to solve for the x
[00:22:36] try to begin it

Guest (Customer)
[00:22:41] alright

Blythe L (Tutor)
[00:23:48] wait
[00:24:01] where did the 3 go?
[00:24:18] okay
[00:24:24] now remember the (x-12)^2
[00:24:35] is equal to (x-12)(x-12)
[00:24:39] you have to FOIL
[00:24:45] it's not the same as x-144
[00:24:58] that's not what it's equal to when you FOIL it out
[00:25:34] so, separate out the (x-12)^2 into (x-12)(x-12)
[00:25:39] and them multiply it out
[00:27:02] you need to multiply -12 times x and -12 times x
[00:27:04] in the middle
[00:27:13] it will be x^2 -12x-12x -144

Guest (Customer)
[00:29:20] is that correct so far

Blythe L (Tutor)
[00:29:39] it needs to be -10x -10x
[00:29:51] you left out one of them

Guest (Customer)
[00:30:28] i dont understand that part

Blythe L (Tutor)
[00:30:47] do you know what the FOIL method is?

Guest (Customer)
[00:31:11] kind of

Amanda S (Tutor)
[00:36:38] Hello,

Guest (Customer)
[00:36:52] hi

Amanda S (Tutor)
[00:36:58] Give me one moment to check over your progress so far :)
[00:38:43] Okay,
[00:39:15] So it seems at this point you're trying to foil (x-12)^2 is this correct?

Guest (Customer)
[00:39:27] yes

Amanda S (Tutor)
[00:39:57] Alright. And you may have noticed you get x times -12 twice when you do this,
[00:40:12] So you'd end up with x^2 -12x -12x - 144
[00:40:28] Does that make sense?

Guest (Customer)
[00:40:33] yes

Amanda S (Tutor)
[00:41:02] Alright

Guest (Customer)
[00:42:34] ok

Amanda S (Tutor)
[00:43:06] Sorry. I'm just checking the original work to see if there's another method, just as this one seems ... a bit long?
[00:43:11] But,
[00:43:18] Continuing with this method,
[00:43:37] If you have x^2 -12x -12x
[00:43:44] and I'm sorry, I made a typo.
[00:43:54] The last term should be plus... +144
[00:44:04] -12 times -12 is positive, +144

Guest (Customer)
[00:44:27] alright

Amanda S (Tutor)
[00:44:40] I'm really sorry. Would it be okay if I get a new board? :(
[00:44:54] I can't see what I'm writing on this one

Guest (Customer)
[00:44:54] yes thats fine

Amanda S (Tutor)
[00:45:07] :-) Thanks

Guest (Customer)
[00:45:22] welcome

Amanda S (Tutor)
[00:46:46] So at this point we were dealing with (x-12)^2 and I was mentioning we'd end up with
[00:46:54] x^2 -12x -12x + 144

Guest (Customer)
[00:47:09] yes

Amanda S (Tutor)
[00:47:34] -12x -12x would combine to give us -24x

Guest (Customer)
[00:48:10] yes

Amanda S (Tutor)
[00:48:26] At this point, I think I'd really want to just get rid of the fractions.
[00:48:54] The way to do that is to multiply everything by the least common denominator.

Guest (Customer)
[00:49:46] 4
[00:49:48] ?

Amanda S (Tutor)
[00:50:24] That would indeed be a common factor, but for a common denominator we want a common multiple
[00:50:59] :( Does that make sense? I know people usually don't stress the difference between those

Guest (Customer)
[00:51:12] yes

Amanda S (Tutor)
[00:51:20] Ok,
[00:51:28] For instance
[00:51:53] The method I sometimes use is a bit counter-intuitive,
[00:52:38] Both the terms have a 4
[00:52:55] But one lacks a six
[00:53:06] And the other lacks 25

Guest (Customer)
[00:53:23] 600

Amanda S (Tutor)
[00:53:28] Right!!
[00:53:45] So from this I get 600 as the lowest common denominator.
[00:53:54] :S As I said, it's not a method most people use.
[00:54:10] But to cancel the fractions, we times each thing by 600.
[00:54:29] Does that reasoning work out for you?

Guest (Customer)
[00:54:37] yes

Amanda S (Tutor)
[00:54:45] Okay
[00:55:04] So that first part is times 600,
[00:55:24] +3 times 600
[00:55:31] and on the right, everything times 600.
[00:55:45] does that make sense?

Guest (Customer)
[00:55:57] yes

Amanda S (Tutor)
[00:56:06] Alright
[00:56:09] On the left side
[00:56:31] If you take 600 times .. well, all the stuff on the left.
[00:56:37] How would you work through tat?
[00:56:40] that*?
[00:57:20] Okay.
[00:57:23] Not quite,
[00:57:34] We'd really think of it as 600 over 1,
[00:58:07] So we'd really be canceling the 600 with the 100
[00:59:29] Okay. For the most part you're getting the idea,
[00:59:41] Let me help with just one or two things here
[01:00:02] For instance you did cancel 600 and 100 out, but
[01:00:16] You still need to cross out the 00's there,
[01:00:26] So that's just 6
[01:00:58] Those parentheses in the first part are still important,
[01:01:18] And one last thing
[01:01:34] When you canceled 600 with -1/24
[01:01:58] So that's 600 times -1/24
[01:02:21] What's 600 divided by -24, what would that give you?

Guest (Customer)
[01:02:47] .25
[01:02:52] -25

Amanda S (Tutor)
[01:02:58] Right,
[01:03:19] So in that whole part there, do you see where I was getting those numbers?

Guest (Customer)
[01:03:35] yes

Amanda S (Tutor)
[01:03:42] Okay.
[01:03:47] At this point
[01:04:05] We have 6(-9(x-10)^2) + 1800
[01:04:29] =
[01:04:41] -25(x^2-24x+144)
[01:05:23] Still a fairly involved problem.
[01:05:38] What do you think the next step would be at this point?

Guest (Customer)
[01:05:53] solve

Amanda S (Tutor)
[01:06:09] Right,
[01:06:27] You may consider simplifying first by taking 6 times -9 in the first part.
[01:07:00] Right,
[01:07:06] And the rest can be left the same for now,
[01:07:14] Unless you'd like to do two steps at once

Guest (Customer)
[01:07:16] ok

Amanda S (Tutor)
[01:07:24] And proceed by foiling (x-10)^2 ?
[01:08:54] Alright
[01:09:15] Though I think you might find you get x^2 -10x and then another -19x
[01:09:19] *-10x sorry
[01:09:30] :/
[01:09:40] x^2 -10x - 10x
[01:09:52] and then -10 times -10 would give you
[01:10:00] positive +100
[01:11:27] I'll get a new whiteboard, :< Sorry! We ran out of room!!
[01:11:41] Do you have any questions so far?

Guest (Customer)
[01:11:40] alright
[01:11:55] no

Amanda S (Tutor)
[01:14:13] Okay, I believe that's what we have so far.
[01:14:16] I'm double checking.
[01:14:33] Could you double check as well to make sure that's the equation as we have it so far?

Guest (Customer)
[01:14:52] yes

Amanda S (Tutor)
[01:15:22] Alright; I believe everything so far should work out.
[01:15:34] What do you think the next step should be?

Guest (Customer)
[01:16:03] combine like terms

Amanda S (Tutor)
[01:16:11] Right,
[01:16:13] :)
[01:17:25] What would you do next?

Guest (Customer)
[01:17:44] get number to one ide
[01:17:47] side

Amanda S (Tutor)
[01:18:04] I can see what you mean, I think,
[01:18:20] But moving the +1800 around doesn't do much? If that's what you mean?

Guest (Customer)
[01:18:44] i can subtract the 144

Amanda S (Tutor)
[01:18:57] Okay. Actually you cannot,
[01:19:04] Because it's in parentheses.
[01:19:08] That's the only reason,

Guest (Customer)
[01:19:11] oh

Amanda S (Tutor)
[01:19:33] It's because it's actually times the -25 that's out front.
[01:20:02] Unless you take that -25 into account, you can't really move the -144, I don't know if that makes sense?

Guest (Customer)
[01:20:19] ok
[01:20:48] so would i have to time the -25 to the whole equation to get rid of it

Amanda S (Tutor)
[01:20:56] Yes;
[01:20:58] In fact,
[01:21:13] At this point it would be great if we could just get rid of any parentheses.
[01:21:18] This means
[01:21:32] On the right we take -25 times each thing it's next to in parentheses,
[01:21:48] And on the left side as well
[01:22:11] Continue,
[01:22:47] Right,
[01:23:20] So on the left side?
[01:24:16] Good!
[01:24:21] One thing on signs
[01:24:39] Negative 54, negative 20
[01:25:07] Okay!
[01:25:24] At this point we have
[01:25:48] -54x^2+1080x -5400
[01:25:49] =
[01:26:19] -25x^2-600x-3600
[01:26:54] *sorry missed the +1800
[01:28:05] :( I'm sorry this problem is taking so long,
[01:28:14] Does this correspond with what you have so far?

Guest (Customer)
[01:28:43] yes

Amanda S (Tutor)
[01:28:51] Okay, great!
[01:28:58] What would you do at this point?

Guest (Customer)
[01:29:39] subtract numebrs from sides
[01:29:45] or add numbers

Amanda S (Tutor)
[01:29:57] Okay,
[01:30:20] Looking at the problem at this point. I'd agree, one one condition.
[01:30:27] We want to get everything on the left side.
[01:30:46] So we may start with -5400+1800 on the right
[01:31:04] Right
[01:31:16] Then looking on the right side, what do you see?

Guest (Customer)
[01:31:37] they cancel out

Amanda S (Tutor)
[01:31:41] :D Right
[01:31:48] :)
[01:32:12] How would you work with the problem now?

Guest (Customer)
[01:32:46] i don't know i am stuck

Amanda S (Tutor)
[01:33:04] Well let's continue with trying to get things on the left side.
[01:33:14] You have -25x^2 on the right

Guest (Customer)
[01:33:35] add 600x

Amanda S (Tutor)
[01:33:48] Sure, that's a good start!
[01:36:07] Okay. So, at this point you have 1680 = 29x^2
[01:36:11] Yes?
[01:36:27] Wait, hang on

Guest (Customer)
[01:36:26] square root

Amanda S (Tutor)
[01:36:40] Yes but I'm missing something.
[01:36:51] Oh ! wait,
[01:37:00] That's still 1680 c
[01:37:05] 1680x*
[01:37:17] Right?

Guest (Customer)
[01:37:26] oh yes

Amanda S (Tutor)
[01:37:49] So we have... 1680x = 29x^2
[01:37:54] Right?

Guest (Customer)
[01:37:59] yes

Amanda S (Tutor)
[01:38:06] Ok
[01:38:12] I'm going to get a new board,

Guest (Customer)
[01:38:45] ok

Amanda S (Tutor)
[01:39:07] One thing I'm thinking at this piont,
[01:39:09] point*
[01:39:55] And just see if this maybe makes sense
[01:40:01] We move everything to one side,

Guest (Customer)
[01:40:10] kind of

Amanda S (Tutor)
[01:40:24] Now both of these has an 'x' term in it
[01:40:38] So for factoring,
[01:40:55] We can take one x out
[01:41:25] Does that kind of make sense?

Guest (Customer)
[01:41:31] yes

Amanda S (Tutor)
[01:41:55] So one x-term out of 29x^2 is what?

Guest (Customer)
[01:42:02] 29x

Amanda S (Tutor)
[01:42:10] RighT!
[01:42:30] How about an x-term out of -1680x ?

Guest (Customer)
[01:42:46] 1680

Amanda S (Tutor)
[01:42:53] Right,
[01:43:07] and so last,
[01:43:12] what do you think we'd do?

Guest (Customer)
[01:43:26] check it

Amanda S (Tutor)
[01:43:44] Well right,
[01:43:47] But to get solutions
[01:43:51] we'd sed
[01:43:55] set*
[01:43:57] x=0
[01:44:07] and 29x-1680 =0
[01:44:48] Does that sound familiar?
[01:45:26] :) Right.
[01:45:32] So that should be your answer,
[01:45:56] and I know that was a super, mega number of steps.

Guest (Customer)
[01:45:56] ok thanks

Amanda S (Tutor)
[01:46:12] Any questions on what we've worked through?

Guest (Customer)
[01:46:17] yeah and no

Amanda S (Tutor)
[01:46:40] Wow, okay! :) Well thanks for holding on there! You did a super job.
[01:46:55] Thanks for using Tutor.com ! Please fill out the survey at the end, if you have time. :) Your feedback really helps!

Guest (Customer)
[01:46:59] thank you for helping me finish and understand

Amanda S (Tutor)
[01:47:10] :) You're very welcome.
[01:47:12] Bye!!