1 00:00:00,000 --> 00:00:00,690 2 00:00:00,690 --> 00:00:04,679 We're on problem 32. 3 00:00:04,679 --> 00:00:10,649 What are the solutions to the equation 1 plus 1 over x 4 00:00:10,650 --> 00:00:15,080 squared is equal to 3 over x? 5 00:00:15,080 --> 00:00:17,309 So at first. This looks like a pretty daunting equation. 6 00:00:17,309 --> 00:00:19,339 You have these x's in the denominator and x squared in 7 00:00:19,339 --> 00:00:20,239 the denominator. 8 00:00:20,239 --> 00:00:23,019 But I think we can simplify it if we can just get rid of 9 00:00:23,019 --> 00:00:24,250 these x squares in the denominator. 10 00:00:24,250 --> 00:00:25,769 The easiest way to do that is to multiply 11 00:00:25,769 --> 00:00:27,530 everything by x squared. 12 00:00:27,530 --> 00:00:30,050 So let's multiply both sides of this equation by x squared. 13 00:00:30,050 --> 00:00:34,340 Then we'll get x squared times 1 is x squared, x squared 14 00:00:34,340 --> 00:00:38,780 times 1 over x squared, that's just 1. 15 00:00:38,780 --> 00:00:42,890 Then x squared times 3 over x, that's 3x squared over x. 16 00:00:42,890 --> 00:00:47,350 x squared divided by x is just x, so that is equal to 3x. 17 00:00:47,350 --> 00:00:51,700 We can subtract 3x from both sides and you get x squared 18 00:00:51,700 --> 00:00:55,450 minus 3x plus 1 is equal to 0. 19 00:00:55,450 --> 00:00:57,810 This is a simple quadratic. 20 00:00:57,810 --> 00:01:00,250 It's not obvious that you can factor it. 21 00:01:00,250 --> 00:01:03,689 In fact, two numbers when you multiply them equal 1, and 22 00:01:03,689 --> 00:01:07,039 then when you add them equal minus 3. 23 00:01:07,040 --> 00:01:08,670 I'm guessing it might even be imaginary. 24 00:01:08,670 --> 00:01:10,109 It will probably not be imaginary, but it's just a 25 00:01:10,109 --> 00:01:10,900 strange number. 26 00:01:10,900 --> 00:01:12,230 So let's use the quadratic equation. 27 00:01:12,230 --> 00:01:14,260 When in doubt, use the quadratic equation. 28 00:01:14,260 --> 00:01:17,750 So minus B, this is B right? 29 00:01:17,750 --> 00:01:21,900 B is this 3 right there, negative 3. 30 00:01:21,900 --> 00:01:22,690 B is negative 3. 31 00:01:22,689 --> 00:01:26,480 So minus B is going to be plus 3, plus or minus the square 32 00:01:26,480 --> 00:01:27,240 root of B squared. 33 00:01:27,239 --> 00:01:33,939 Minus 3 squared is 9, minus 4 times A, which is 1, times C 34 00:01:33,939 --> 00:01:34,480 which is 1. 35 00:01:34,480 --> 00:01:38,060 So it's minus 4, all of that over 2A. 36 00:01:38,060 --> 00:01:40,740 A is 1 so it's just over 2. 37 00:01:40,739 --> 00:01:47,079 That is equal to 3/2 plus or minus the square 38 00:01:47,079 --> 00:01:49,129 root of 5 over 2. 39 00:01:49,129 --> 00:01:50,989 I just separated these out because I'm looking at the 40 00:01:50,989 --> 00:01:52,390 choices and it seems like they did that. 41 00:01:52,390 --> 00:01:55,769 So you could've said that's 3/2 plus square root of 5 over 42 00:01:55,769 --> 00:02:00,390 2 or 3/2 minus the square root of 5 over 2. 43 00:02:00,390 --> 00:02:01,859 I just did that because it seems like that's 44 00:02:01,859 --> 00:02:03,250 how they write it. 45 00:02:03,250 --> 00:02:05,840 That is choice A. 46 00:02:05,840 --> 00:02:10,870 Next problem, 33. 47 00:02:10,870 --> 00:02:13,840 I think this one actually might be good to copy and 48 00:02:13,840 --> 00:02:15,509 paste the problem. 49 00:02:15,509 --> 00:02:18,229 Let me see if I can do this. 50 00:02:18,229 --> 00:02:21,329 OK, there are two numbers with the following properties. 51 00:02:21,330 --> 00:02:23,450 Let me write down the properties and let me copy and 52 00:02:23,449 --> 00:02:25,719 paste it for you. 53 00:02:25,719 --> 00:02:27,409 OK, I've copied it. 54 00:02:27,409 --> 00:02:29,349 Let me go here. 55 00:02:29,349 --> 00:02:32,009 Then I've pasted it for you. 56 00:02:32,009 --> 00:02:33,259 All right. 57 00:02:33,259 --> 00:02:36,949 58 00:02:36,949 --> 00:02:38,369 So there's two numbers with the following properties. 59 00:02:38,370 --> 00:02:40,740 The second number is 3 more than the first number. 60 00:02:40,740 --> 00:02:44,180 So let's say S for second number and F for first number. 61 00:02:44,180 --> 00:02:48,490 So the second number is 3 more than the first number. 62 00:02:48,490 --> 00:02:52,680 So the second number is equal to the first number plus 3. 63 00:02:52,680 --> 00:02:54,569 That's from statement 1. 64 00:02:54,569 --> 00:02:57,949 The product of the two numbers is 9 more than the sum. 65 00:02:57,949 --> 00:03:07,449 So the product of the two numbers, that's S times F is 9 66 00:03:07,449 --> 00:03:15,289 more, 9 plus their sum, plus S plus F. 67 00:03:15,289 --> 00:03:17,179 So let's see, we have two equations and two unknowns. 68 00:03:17,180 --> 00:03:18,379 This is nonlinear because I'm 69 00:03:18,379 --> 00:03:20,000 multiplying these two variables. 70 00:03:20,000 --> 00:03:23,069 But I think we should be able to solve them 71 00:03:23,069 --> 00:03:24,689 one way or the other. 72 00:03:24,689 --> 00:03:27,729 So let's see, we have what S is equal to. 73 00:03:27,729 --> 00:03:31,439 So let's just substitute that back into this equation. 74 00:03:31,439 --> 00:03:34,349 So let's say that S is equal to F plus 3. 75 00:03:34,349 --> 00:03:41,340 So if we substitute for these S's, we get F plus 3 times F 76 00:03:41,340 --> 00:03:46,990 is equal to 9 plus F plus 3, right, instead of an S, F plus 77 00:03:46,990 --> 00:03:49,772 3 and then plus F. 78 00:03:49,771 --> 00:03:51,449 Let's see if we can simplify this. 79 00:03:51,449 --> 00:03:58,459 F times F is F squared plus 3F is equal to 9 plus 80 00:03:58,460 --> 00:04:05,730 3 is 12 plus 2F. 81 00:04:05,729 --> 00:04:09,739 Subtract 2F from both sides, you get F squared plus F is 82 00:04:09,740 --> 00:04:11,180 equal to 12. 83 00:04:11,180 --> 00:04:18,000 Subtract 12, you get F squared plus F minus 12 is equal to 0. 84 00:04:18,000 --> 00:04:19,379 This one looks factorable. 85 00:04:19,379 --> 00:04:21,509 I don't have to take out the quadratic equation. 86 00:04:21,509 --> 00:04:30,170 Let's see, this is F plus 4 times F minus 3, right? 87 00:04:30,170 --> 00:04:31,509 Because when you multiply those, you get negative 12. 88 00:04:31,509 --> 00:04:35,399 When you add those, you get plus 1, so that is equal to 0. 89 00:04:35,399 --> 00:04:38,419 So in order for this to be true, one or both of these 90 00:04:38,420 --> 00:04:39,500 have to be equal to 0. 91 00:04:39,500 --> 00:04:45,870 So if F plus 4 is equal to 0, that means F could be 92 00:04:45,870 --> 00:04:47,519 equal to minus 4. 93 00:04:47,519 --> 00:04:50,500 If F minus 3 is equal to 0, then that says 94 00:04:50,500 --> 00:04:51,689 that F could be 3. 95 00:04:51,689 --> 00:04:54,810 So F could be minus 4 or 3. 96 00:04:54,810 --> 00:04:59,709 Now, S is F plus 3, so if we're dealing with the minus 4 97 00:04:59,709 --> 00:05:03,729 scenario, if F is equal to minus 4, then what is S? 98 00:05:03,730 --> 00:05:07,939 Then S is going to be minus 4 plus 3, and S is going to be 99 00:05:07,939 --> 00:05:09,389 equal to minus 1. 100 00:05:09,389 --> 00:05:13,089 Then if F is equal to 3, than S is equal to 6. 101 00:05:13,089 --> 00:05:16,009 So let's see if we see either of these combinations. 102 00:05:16,009 --> 00:05:19,899 Minus 4, minus 1, that's choice B. 103 00:05:19,899 --> 00:05:21,039 Excellent. 104 00:05:21,040 --> 00:05:27,890 All right, problem 34. 105 00:05:27,889 --> 00:05:31,399 Let me see, maybe I should copy and paste these word 106 00:05:31,399 --> 00:05:35,459 problems so we can see how we parse the problems. So I've 107 00:05:35,459 --> 00:05:41,319 copied it, then I'll go here, and then pasted it. 108 00:05:41,319 --> 00:05:45,019 Jenny is solving the equation x squared minus 8x equals 9 by 109 00:05:45,019 --> 00:05:46,469 completing the square. 110 00:05:46,470 --> 00:05:48,910 What number should be added to both sides of the equation to 111 00:05:48,910 --> 00:05:49,660 complete the square? 112 00:05:49,660 --> 00:05:55,280 So x squared minus 8x is equal to 9. 113 00:05:55,279 --> 00:05:57,599 I wrote it with space for a reason. 114 00:05:57,600 --> 00:05:59,420 When you're completing the square, you're trying to turn 115 00:05:59,420 --> 00:06:02,699 the left-hand side of this equation into some type of a 116 00:06:02,699 --> 00:06:04,649 perfect square. 117 00:06:04,649 --> 00:06:07,589 So if it's a perfect square, I have two numbers. 118 00:06:07,589 --> 00:06:11,879 It's the same number that when you add them together, you get 119 00:06:11,879 --> 00:06:14,810 minus 8, and when you square them, you should get something 120 00:06:14,810 --> 00:06:15,470 else, right? 121 00:06:15,470 --> 00:06:17,410 So what's half of minus 8? 122 00:06:17,410 --> 00:06:21,270 Half of minus 8 is minus 4. 123 00:06:21,269 --> 00:06:24,430 So minus 4 squared is 16. 124 00:06:24,430 --> 00:06:27,829 So if I add 16 to both sides, I'm all set. 125 00:06:27,829 --> 00:06:28,469 Why did that work? 126 00:06:28,470 --> 00:06:29,460 Well, now it's a perfect square. 127 00:06:29,459 --> 00:06:34,939 This is now x minus 4 squared is equal to 9 plus 16 is 25. 128 00:06:34,939 --> 00:06:36,399 They're not even asking us to solve it. 129 00:06:36,399 --> 00:06:39,069 They just want to know what we had to add to both sides. 130 00:06:39,069 --> 00:06:40,990 So it's 16, D. 131 00:06:40,990 --> 00:06:43,310 Remember the whole logic here, and I've done a few videos on 132 00:06:43,310 --> 00:06:46,030 completing the squares, is what number do I add here to 133 00:06:46,029 --> 00:06:47,979 make this a perfect square? 134 00:06:47,980 --> 00:06:51,470 You say, OK, I have a minus 8x, so I take half of this 135 00:06:51,470 --> 00:06:54,540 number, because the same number added to itself twice 136 00:06:54,540 --> 00:06:56,590 is going to become minus 8. 137 00:06:56,589 --> 00:06:58,375 I take half of that number, then I squared it. 138 00:06:58,375 --> 00:07:00,120 So half of minus 8 is minus 4. 139 00:07:00,120 --> 00:07:01,990 If you square it, you get the 16. 140 00:07:01,990 --> 00:07:03,420 So add 16 to both sides, you get this. 141 00:07:03,420 --> 00:07:04,420 You can actually solve for this. 142 00:07:04,420 --> 00:07:06,069 x minus 4 is plus or minus 5. 143 00:07:06,069 --> 00:07:07,110 You keep going. 144 00:07:07,110 --> 00:07:10,069 That's actually where the quadratic equation comes from. 145 00:07:10,069 --> 00:07:13,139 Anyway, next problem. 146 00:07:13,139 --> 00:07:20,979 16 was choice number D. 147 00:07:20,980 --> 00:07:25,950 I'm going to copy and paste this entire problem here. 148 00:07:25,949 --> 00:07:29,810 Let's go here and paste it here. 149 00:07:29,810 --> 00:07:32,660 OK, which of the following most accurately describes the 150 00:07:32,660 --> 00:07:36,810 translation of the graph y is equal to x plus 3 squared 151 00:07:36,810 --> 00:07:41,555 minus 2 to the graph y equals x minus 2 squared plus 2? 152 00:07:41,555 --> 00:07:44,069 153 00:07:44,069 --> 00:07:50,089 So the y translation tends to be pretty easy to figure out. 154 00:07:50,089 --> 00:07:52,609 Let me just draw some example graph. 155 00:07:52,610 --> 00:07:55,650 156 00:07:55,649 --> 00:07:59,189 So if I had the graph x squared, the graph x squared 157 00:07:59,189 --> 00:08:00,540 looks something like this. 158 00:08:00,540 --> 00:08:01,840 Let's see if I can draw it. 159 00:08:01,839 --> 00:08:04,979 The graph x squared looks something like this, right? 160 00:08:04,980 --> 00:08:05,790 And it intersects. 161 00:08:05,790 --> 00:08:08,330 When x is equal to 0, we're at our minimum point. 162 00:08:08,329 --> 00:08:11,209 And any other value increases in both directions. 163 00:08:11,209 --> 00:08:16,149 The graph of x squared plus 2, you're shifting up. 164 00:08:16,149 --> 00:08:17,870 This is the graph of x squared plus 2. 165 00:08:17,870 --> 00:08:20,030 You would shift it up by 2. 166 00:08:20,029 --> 00:08:24,419 The graph of x squared minus 2, you would shift down by 2. 167 00:08:24,420 --> 00:08:27,210 This would be x squared plus 2. 168 00:08:27,209 --> 00:08:30,500 This would be x squared minus 2. 169 00:08:30,500 --> 00:08:34,110 So the shift in the y direction is very easy to see. 170 00:08:34,110 --> 00:08:44,759 So if we're going from something minus 2 to plus 2 171 00:08:44,759 --> 00:08:48,110 we're going to be shifting it up 4, right? 172 00:08:48,110 --> 00:08:49,710 So that's always the easy one to just 173 00:08:49,710 --> 00:08:51,120 eyeball and figure out. 174 00:08:51,120 --> 00:08:53,919 So we're definitely going to be shifting from minus 2 to 2. 175 00:08:53,919 --> 00:08:54,860 So it's up 4. 176 00:08:54,860 --> 00:08:58,519 It's either going to be choice A or choice D. 177 00:08:58,519 --> 00:09:02,269 The left/right shift is often a little bit more hard for 178 00:09:02,269 --> 00:09:06,399 people to visualize or to at least internalize, but I'll 179 00:09:06,399 --> 00:09:09,240 give you an attempt. 180 00:09:09,240 --> 00:09:10,100 Let's just go back to this. 181 00:09:10,100 --> 00:09:12,300 This is the graph of x squared, this yellow line 182 00:09:12,299 --> 00:09:14,919 right there. 183 00:09:14,919 --> 00:09:16,279 That's the graph of x squared. 184 00:09:16,279 --> 00:09:17,120 Let me ask you a question. 185 00:09:17,120 --> 00:09:23,820 What is the graph of x minus 3 squared? 186 00:09:23,820 --> 00:09:28,040 So does this shift it down 3 to the negative direction or 3 187 00:09:28,039 --> 00:09:28,659 to the positive direction? 188 00:09:28,659 --> 00:09:31,289 Your intuition might say, oh, I'm subtracting 3. 189 00:09:31,289 --> 00:09:33,129 When I did minus 2, I shifted down. 190 00:09:33,129 --> 00:09:34,840 But it's actually the opposite here. 191 00:09:34,840 --> 00:09:37,230 Because you have to think about for what value of x am I 192 00:09:37,230 --> 00:09:40,539 going to have a 0 squared here? 193 00:09:40,539 --> 00:09:43,099 That happens with x is equal to 3. 194 00:09:43,100 --> 00:09:44,200 So you can think of it this way. 195 00:09:44,200 --> 00:09:48,190 Now, when we're at this point, when x is equal to 3, it's the 196 00:09:48,190 --> 00:09:51,650 same thing as this point when we have just x squared. 197 00:09:51,649 --> 00:09:55,480 When you put 3 in here, this whole expression becomes zero. 198 00:09:55,480 --> 00:09:58,610 As you get above 3, that's like going above zero. 199 00:09:58,610 --> 00:10:01,370 As you go below 3, that's like going below zero. 200 00:10:01,370 --> 00:10:07,090 So this graph will just get shifted to the right by 3. 201 00:10:07,090 --> 00:10:10,242 That's x minus 3 shifts to the right by 3. 202 00:10:10,241 --> 00:10:14,409 x plus three would go in the other direction, because when 203 00:10:14,409 --> 00:10:16,870 x is minus 3, that's when it would equal to zero. 204 00:10:16,870 --> 00:10:17,580 I haven't written that down. 205 00:10:17,580 --> 00:10:18,400 So let's think about this. 206 00:10:18,399 --> 00:10:25,639 We're going from x plus 3, so if this is x squared, x plus 3 207 00:10:25,639 --> 00:10:28,100 would look something-- let me do it in a different color. x 208 00:10:28,100 --> 00:10:30,409 plus 3 is actually shifted to the left. 209 00:10:30,409 --> 00:10:31,629 The way I always think about it, there's two ways 210 00:10:31,629 --> 00:10:32,990 to think about it. 211 00:10:32,990 --> 00:10:36,389 The y shift is intuitive and the x shift might not be. 212 00:10:36,389 --> 00:10:38,509 If you have a plus 3 here, you're actually shifting in 213 00:10:38,509 --> 00:10:40,069 the downward direction. 214 00:10:40,070 --> 00:10:43,430 The way to actually think about the intuition is when 215 00:10:43,429 --> 00:10:45,569 will this whole expression equal 0? 216 00:10:45,570 --> 00:10:49,740 This whole expression equals 0 when x is equal to minus 3. 217 00:10:49,740 --> 00:10:53,629 So that's the point at which you're getting 0 squared. 218 00:10:53,629 --> 00:10:55,080 When I'm drawing these graphs, I'm not 219 00:10:55,080 --> 00:10:56,770 doing the y shift here. 220 00:10:56,769 --> 00:11:00,350 So this is going to be shifted to the left 3. 221 00:11:00,350 --> 00:11:03,250 This is going to be shifted to the right by 2. 222 00:11:03,250 --> 00:11:05,809 So if this is shifted to the left 3 and this is shifted to 223 00:11:05,809 --> 00:11:08,699 the right by 2, to go from this to this, you're shifting 224 00:11:08,700 --> 00:11:09,950 to the right by 5. 225 00:11:09,950 --> 00:11:14,180 226 00:11:14,179 --> 00:11:17,000 So the actual graph x plus 3 squared minus 2 227 00:11:17,000 --> 00:11:20,500 is going to be here. 228 00:11:20,500 --> 00:11:23,710 Then to go here, you have a plus 2, so you're shifting the 229 00:11:23,710 --> 00:11:27,970 graph up by 4, and then you're going to x minus 2. 230 00:11:27,970 --> 00:11:31,100 So this graph right here is going to be up here. 231 00:11:31,100 --> 00:11:35,550 So you're shifting up by 4 and then you're shifting to the 232 00:11:35,549 --> 00:11:38,639 right by 5. 233 00:11:38,639 --> 00:11:41,179 Actually, even if you're confused with your shifting 234 00:11:41,179 --> 00:11:42,739 left or right, you can just say the difference between 235 00:11:42,740 --> 00:11:45,110 plus 3 and minus 2 is 5, and 5 is only there. 236 00:11:45,110 --> 00:11:47,440 But you should hopefully understand the problems a 237 00:11:47,440 --> 00:11:48,800 little bit deeper than that. 238 00:11:48,799 --> 00:11:51,399 Anyway, I'll see you in the next video. 239 00:11:51,399 --> 00:11:53,000