1 00:00:00,000 --> 00:00:00,900 2 00:00:00,900 --> 00:00:02,279 We're on problem 48. 3 00:00:02,279 --> 00:00:05,799 It says if x squared is added to x, the sum is 42. 4 00:00:05,799 --> 00:00:07,219 So let's just write that out. 5 00:00:07,219 --> 00:00:13,390 If x squared is added to x, the sum is equal to 42. 6 00:00:13,390 --> 00:00:16,079 Which of the following could be the value of x? 7 00:00:16,079 --> 00:00:17,660 So essentially they just want us to solve this equation. 8 00:00:17,660 --> 00:00:22,929 So the easiest way to do it is to write it as a quadratic 9 00:00:22,929 --> 00:00:24,500 equaling 0 and then factoring it. 10 00:00:24,500 --> 00:00:30,129 So we could write this as x squared plus x minus 42 is 11 00:00:30,129 --> 00:00:31,559 equal to 0. 12 00:00:31,559 --> 00:00:32,019 And let's think. 13 00:00:32,020 --> 00:00:36,040 What two numbers when I add them equal 1, and when I 14 00:00:36,039 --> 00:00:38,710 multiply them equal minus 42? 15 00:00:38,710 --> 00:00:42,109 And the fact that when I multiply them equals minus 42 16 00:00:42,109 --> 00:00:44,530 tells me that one of them has to be positive and one of them 17 00:00:44,530 --> 00:00:45,289 has to be negative. 18 00:00:45,289 --> 00:00:47,280 That's the only way that when you multiply two numbers 19 00:00:47,280 --> 00:00:48,700 you're going to have a negative number. 20 00:00:48,700 --> 00:00:50,500 So one of them has to be positive, one of them has to 21 00:00:50,500 --> 00:00:51,369 be negative. 22 00:00:51,369 --> 00:00:53,640 And so when we're adding a positive and a negative, 23 00:00:53,640 --> 00:00:55,429 you're really finding the difference between the two. 24 00:00:55,429 --> 00:00:58,079 So the difference between the two numbers has to be 1 and 25 00:00:58,079 --> 00:01:00,000 their product has to be 42. 26 00:01:00,000 --> 00:01:01,609 And I noticed when I see 42 I immediately 27 00:01:01,609 --> 00:01:02,769 think, oh, 6 and 7. 28 00:01:02,770 --> 00:01:04,930 6 times 7 is 42. 29 00:01:04,930 --> 00:01:07,850 And since when you add them you get a positive 1, 7's 30 00:01:07,849 --> 00:01:11,780 probably the positive one and minus 6 or 6 is probably the 31 00:01:11,780 --> 00:01:12,730 negative one. 32 00:01:12,730 --> 00:01:14,210 So let's try it out. 33 00:01:14,209 --> 00:01:21,189 x plus 7 times x minus 6 equal to 0. 34 00:01:21,189 --> 00:01:26,459 And right, 7 times minus 6 is minus 42. 35 00:01:26,459 --> 00:01:30,459 It's really 7x plus minus 6x is equal to positive x. 36 00:01:30,459 --> 00:01:33,739 Or you could think 7 plus minus 6 is equal to the 37 00:01:33,739 --> 00:01:36,465 coefficient on the x, which is 1. 38 00:01:36,465 --> 00:01:37,890 But anyway, that works. 39 00:01:37,890 --> 00:01:40,049 And you could multiply this out and try it out. 40 00:01:40,049 --> 00:01:42,859 And everything I'm saying, it's not some voodoo. 41 00:01:42,859 --> 00:01:46,200 The reason why I say that they have to add up to the 1 is 42 00:01:46,200 --> 00:01:48,620 because when you multiply this out, that's what 43 00:01:48,620 --> 00:01:49,870 builds up this term. 44 00:01:49,870 --> 00:01:54,370 This 7 times x plus the minus 6 times the other x. 45 00:01:54,370 --> 00:01:56,880 That's what builds this term when you multiply it out. 46 00:01:56,879 --> 00:01:59,009 This term comes from the x times the x. 47 00:01:59,010 --> 00:02:02,620 The minus 42 comes from the 7 times the minus 6. 48 00:02:02,620 --> 00:02:04,640 Anyway, now we're at this point. 49 00:02:04,640 --> 00:02:07,209 We say OK, well how do we get-- we have two things when 50 00:02:07,209 --> 00:02:08,840 you multiply them equal to 0. 51 00:02:08,840 --> 00:02:11,120 Well that means that one of them or both of them have to 52 00:02:11,120 --> 00:02:12,110 be equal to 0. 53 00:02:12,110 --> 00:02:16,520 So that means that x plus 7 is equal to 0, which would mean 54 00:02:16,520 --> 00:02:17,890 subtract 7 from both sides. 55 00:02:17,889 --> 00:02:20,289 Which means x is equal to minus 7. 56 00:02:20,289 --> 00:02:23,579 Or x minus 6 is equal to 0. 57 00:02:23,580 --> 00:02:26,090 Add 6 to both sides. x is equal to 6. x would 58 00:02:26,090 --> 00:02:28,110 be 6 or minus 7. 59 00:02:28,110 --> 00:02:29,830 And they have one of the choices there, 60 00:02:29,830 --> 00:02:32,710 which was choice A. 61 00:02:32,710 --> 00:02:34,340 Next problem. 62 00:02:34,340 --> 00:02:35,819 49. 63 00:02:35,819 --> 00:02:38,739 What quantity should be added to both sides of this equation 64 00:02:38,740 --> 00:02:40,320 to complete the square? 65 00:02:40,319 --> 00:02:41,930 So when you complete the square you want the thing to 66 00:02:41,930 --> 00:02:44,740 just look like a-- you want whatever's on the left-hand 67 00:02:44,740 --> 00:02:46,189 side to look like a perfect square. 68 00:02:46,189 --> 00:02:47,349 And what do I mean by perfect square? 69 00:02:47,349 --> 00:02:54,579 So if I had x plus a squared, that's equal to x plus a 70 00:02:54,580 --> 00:02:57,030 times x plus a. 71 00:02:57,030 --> 00:03:00,370 And that's equal to x times x, x squared. 72 00:03:00,370 --> 00:03:03,890 x times that a, so that's plus ax. 73 00:03:03,889 --> 00:03:05,679 And now this a times this x. 74 00:03:05,680 --> 00:03:07,250 So that's another ax. 75 00:03:07,250 --> 00:03:08,729 Plus this a times that a. 76 00:03:08,729 --> 00:03:10,409 So plus a squared. 77 00:03:10,409 --> 00:03:12,099 And that's equal to x squared. 78 00:03:12,099 --> 00:03:17,719 Plus-- we have two of these now-- plus 2ax plus a squared. 79 00:03:17,719 --> 00:03:21,520 So essentially we want this, we want the left-hand side to 80 00:03:21,520 --> 00:03:22,930 have this form. 81 00:03:22,930 --> 00:03:24,330 So we say, this is a perfect square. 82 00:03:24,330 --> 00:03:26,742 We can say that's the same thing as x plus a squared. 83 00:03:26,741 --> 00:03:29,819 So let's think about how we can do it. 84 00:03:29,819 --> 00:03:37,039 If we have x squared minus 8x is equal to 5 and I put a 85 00:03:37,039 --> 00:03:39,639 space here for a reason because we want to add or 86 00:03:39,639 --> 00:03:42,489 subtract something here so it looks like a perfect square. 87 00:03:42,490 --> 00:03:44,530 So think about it. 88 00:03:44,530 --> 00:03:48,599 When we have this format, in order for this thing to be a 89 00:03:48,599 --> 00:03:53,819 perfect square, whatever this coefficient is right here, 90 00:03:53,819 --> 00:03:57,430 this term right here has to be half of this, squared. 91 00:03:57,430 --> 00:04:00,310 a squared is half of 2a squared. 92 00:04:00,310 --> 00:04:03,990 So if we took half of minus 8, that's minus 4. 93 00:04:03,990 --> 00:04:06,890 In this case, if we said 2a is equal to 8, a 94 00:04:06,889 --> 00:04:08,719 would be minus 4. 95 00:04:08,719 --> 00:04:10,909 And so minus 4 squared is what? 96 00:04:10,909 --> 00:04:14,829 It's plus 16. 97 00:04:14,830 --> 00:04:15,990 And this is an equation. 98 00:04:15,990 --> 00:04:18,004 So one you do to one side of an equation you have to do to 99 00:04:18,004 --> 00:04:18,954 the other side of the equation. 100 00:04:18,954 --> 00:04:21,569 So you have to say that that is also equal to. 101 00:04:21,569 --> 00:04:22,819 So you have to add 16 to both sides. 102 00:04:22,819 --> 00:04:26,599 Otherwise you're changing the equation. 103 00:04:26,600 --> 00:04:28,610 Now this, hopefully you recognize this as already a 104 00:04:28,610 --> 00:04:29,300 perfect square. 105 00:04:29,300 --> 00:04:31,270 I mean you could look at this pattern up here or you could 106 00:04:31,269 --> 00:04:35,089 say, OK, if I add minus 4 to itself twice I get minus 8. 107 00:04:35,089 --> 00:04:37,679 If I multiply it by itself I get 16. 108 00:04:37,680 --> 00:04:41,790 So this is x minus 4 squared. 109 00:04:41,790 --> 00:04:44,490 That's equal to 25. 110 00:04:44,490 --> 00:04:46,370 And actually, just if you're curious-- and we did this in 111 00:04:46,370 --> 00:04:48,170 the Khan Academy, we did a couple of videos on this-- 112 00:04:48,170 --> 00:04:50,400 this is how you prove the quadratic equation. 113 00:04:50,399 --> 00:04:54,049 You essentially complete the square with arbitrary numbers 114 00:04:54,050 --> 00:04:56,319 a, b, and c, and you get the quadratic equation. 115 00:04:56,319 --> 00:04:59,540 You know, we show it in 10 minutes, so it's not this 116 00:04:59,540 --> 00:05:02,270 impossibly hard thing to understand. 117 00:05:02,269 --> 00:05:04,859 They just want to know, what do you add to both sides of 118 00:05:04,860 --> 00:05:06,220 this equation? 119 00:05:06,220 --> 00:05:08,500 What quantity should be added to both sides of this equation 120 00:05:08,500 --> 00:05:09,370 to complete the square? 121 00:05:09,370 --> 00:05:12,100 So the answer to this one was 16. 122 00:05:12,100 --> 00:05:14,670 But they just as equally could have said, solve it by 123 00:05:14,670 --> 00:05:15,860 completing the square. 124 00:05:15,860 --> 00:05:18,730 And you'd say oh, x minus 4 squared is equal to 25. 125 00:05:18,730 --> 00:05:22,160 So x minus 4 is equal to plus or minus 5. 126 00:05:22,160 --> 00:05:24,310 And then you could say, x is equal to plus or 127 00:05:24,310 --> 00:05:26,689 minus 5 plus 4. 128 00:05:26,689 --> 00:05:29,930 And then you could say, OK, which is 4 plus 129 00:05:29,930 --> 00:05:31,189 positive 5 is 9. 130 00:05:31,189 --> 00:05:35,779 4 plus minus 5 is-- or, minus 1. 131 00:05:35,779 --> 00:05:37,159 Anyway, they didn't ask us that, so we don't have to 132 00:05:37,160 --> 00:05:39,350 spend too much time thinking about it. 133 00:05:39,350 --> 00:05:42,430 134 00:05:42,430 --> 00:05:49,310 Let's see, we're on problem 50. 135 00:05:49,310 --> 00:05:53,069 Let me see, problem 50. 136 00:05:53,069 --> 00:06:01,969 I'll copy and paste 50 and 51. 137 00:06:01,970 --> 00:06:05,190 All right, what are the solutions for the quadratic 138 00:06:05,189 --> 00:06:12,189 equation x squared plus 6x is equal to 16? 139 00:06:12,189 --> 00:06:15,350 And the temptation here is really to kind of try to solve 140 00:06:15,350 --> 00:06:16,870 it the way you do a linear equation. 141 00:06:16,870 --> 00:06:19,110 I don't know, factor out an x and-- I don't know, do 142 00:06:19,110 --> 00:06:19,580 whatever else. 143 00:06:19,579 --> 00:06:21,009 But the important thing to recognize is this is a 144 00:06:21,009 --> 00:06:22,219 quadratic equation. 145 00:06:22,220 --> 00:06:24,910 And the easiest way to solve it is to put all the terms on 146 00:06:24,910 --> 00:06:26,660 one side and then get a 0 on the other side. 147 00:06:26,660 --> 00:06:28,640 And then either factor it or use the 148 00:06:28,639 --> 00:06:30,089 actual quadratic equation. 149 00:06:30,089 --> 00:06:32,299 Or complete the square, whatever you need to do. 150 00:06:32,300 --> 00:06:34,970 So let's subtract 16 from both sides. 151 00:06:34,970 --> 00:06:40,230 And you get x squared plus 6x minus 16 is equal to 0. 152 00:06:40,230 --> 00:06:43,270 I just subtracted 16 from both sides to get here. 153 00:06:43,269 --> 00:06:45,599 And before just jumping into the quadratic equation, let's 154 00:06:45,600 --> 00:06:47,270 see if we can factor it by inspection. 155 00:06:47,269 --> 00:06:54,164 So what two numbers, when I add them, equal 6-- and we 156 00:06:54,165 --> 00:06:58,602 want positive 6-- and when I multiply them equal minus 16? 157 00:06:58,601 --> 00:07:01,359 And once again, since it's a minus 16, if you multiply two 158 00:07:01,360 --> 00:07:03,310 numbers you get a negative number. 159 00:07:03,310 --> 00:07:04,530 They have to be different signs. 160 00:07:04,529 --> 00:07:07,559 One has to be positive and one has to be negative. 161 00:07:07,560 --> 00:07:10,780 And their difference will be 6 because one's positive and 162 00:07:10,779 --> 00:07:11,434 one's negative. 163 00:07:11,435 --> 00:07:12,990 So let me think about it. 164 00:07:12,990 --> 00:07:16,470 So if I had minus-- well, 8 and 2 is equal to 16. 165 00:07:16,470 --> 00:07:17,390 And they're 6 apart. 166 00:07:17,389 --> 00:07:19,800 So if I did plus 8 and minus 2-- right. 167 00:07:19,800 --> 00:07:22,370 Plus 8 and minus 2 is positive 6. 168 00:07:22,370 --> 00:07:27,360 So it's x plus 8 times x minus 2. 169 00:07:27,360 --> 00:07:28,620 And that really just takes a lot of practice. 170 00:07:28,620 --> 00:07:29,870 You say, OK, what two numbers? 171 00:07:29,870 --> 00:07:30,879 16. 172 00:07:30,879 --> 00:07:31,339 OK. 173 00:07:31,339 --> 00:07:32,009 8 and 2. 174 00:07:32,009 --> 00:07:34,289 Well they're going to have to be different signs. 175 00:07:34,290 --> 00:07:36,480 But I have a positive one here, so whichever number's 176 00:07:36,480 --> 00:07:38,430 larger is probably going to be the positive one. 177 00:07:38,430 --> 00:07:40,120 So positive 8 and minus 2. 178 00:07:40,120 --> 00:07:41,790 Yeah, when you add them up, they equal minus 6. 179 00:07:41,790 --> 00:07:43,660 Yeah, it works. 180 00:07:43,660 --> 00:07:46,120 So you set that equal to 0. 181 00:07:46,120 --> 00:07:48,910 And you say OK, this has to be equal to 0, or that has to be 182 00:07:48,910 --> 00:07:51,320 equal to 0. 183 00:07:51,319 --> 00:07:54,219 So x is either equal to minus 8. 184 00:07:54,220 --> 00:07:57,830 If you say x plus 8 is equal to 0 then subtract 8 from both 185 00:07:57,829 --> 00:07:59,589 sides, you get x is equal to minus 8. 186 00:07:59,589 --> 00:08:01,074 I shouldn't have skipped that step, but 187 00:08:01,074 --> 00:08:02,029 I'll do the step here. 188 00:08:02,029 --> 00:08:04,529 Or you could say x minus 2 is equal to 0. 189 00:08:04,529 --> 00:08:07,219 Add 2 to both sides, you get x is equal to 2. 190 00:08:07,220 --> 00:08:10,070 What x makes this term equal to 0? 191 00:08:10,069 --> 00:08:11,329 And you could look at it from inspection. 192 00:08:11,329 --> 00:08:19,189 So x could either be minus 8 or 2, and that is choice C. 193 00:08:19,189 --> 00:08:21,290 Problem 51. 194 00:08:21,290 --> 00:08:24,970 Leanne correctly solved the equation x squared plus 4x 195 00:08:24,970 --> 00:08:27,020 equals 6 by completing the square. 196 00:08:27,019 --> 00:08:29,529 Which equation is part of her solution? 197 00:08:29,529 --> 00:08:31,384 OK, so the same thing. 198 00:08:31,384 --> 00:08:34,009 x squared plus 4x. 199 00:08:34,009 --> 00:08:35,259 And when you complete the square you're going to add 200 00:08:35,259 --> 00:08:35,808 something here. 201 00:08:35,808 --> 00:08:36,953 So I'm going to leave a little blank. 202 00:08:36,953 --> 00:08:38,849 Is equal to 6. 203 00:08:38,850 --> 00:08:41,879 So what could I add here that makes this expression look 204 00:08:41,879 --> 00:08:43,950 like a perfect square? 205 00:08:43,950 --> 00:08:45,320 Well, got to the pattern that we did a 206 00:08:45,320 --> 00:08:46,850 couple of problems ago. 207 00:08:46,850 --> 00:08:50,840 Whatever's here should be the square of half of this. 208 00:08:50,840 --> 00:08:53,509 So 4-- well, half of that is 2. 209 00:08:53,509 --> 00:08:54,569 2 squared is 4. 210 00:08:54,570 --> 00:08:56,400 So I should add 4 to that side. 211 00:08:56,399 --> 00:08:57,629 If I add 4 to that side, I have to add 4 to 212 00:08:57,629 --> 00:08:59,490 this side as well. 213 00:08:59,490 --> 00:09:02,080 And now this 2 plus 2 is equal to 4. 214 00:09:02,080 --> 00:09:03,800 2 times 2 is equal to 4. 215 00:09:03,799 --> 00:09:06,339 So this is x plus 2 squared. 216 00:09:06,340 --> 00:09:07,660 And I really want you to get the intuition. 217 00:09:07,659 --> 00:09:09,750 Don't memorize the steps for completing the square. 218 00:09:09,750 --> 00:09:12,379 I want you to really understand why. 219 00:09:12,379 --> 00:09:14,120 This is the square of half of that. 220 00:09:14,120 --> 00:09:15,230 And we showed it in the beginning. 221 00:09:15,230 --> 00:09:18,649 Square a lot of binomials and see for yourself that that's 222 00:09:18,649 --> 00:09:20,269 always going to be the case. 223 00:09:20,269 --> 00:09:22,199 Anyway, so this is x plus 2 squared. 224 00:09:22,200 --> 00:09:25,360 That's going to be equal to-- 6 plus 4 is equal to 10. 225 00:09:25,360 --> 00:09:28,169 And that is choice B. 226 00:09:28,169 --> 00:09:30,841 227 00:09:30,841 --> 00:09:34,759 I think we have time for one more. 228 00:09:34,759 --> 00:09:40,980 One more problem, problem 52. 229 00:09:40,980 --> 00:09:45,330 Copied it and now I have pasted it. 230 00:09:45,330 --> 00:09:47,410 Carter is solving this equation by factoring. 231 00:09:47,409 --> 00:09:51,610 Which expression could be one of his correct factors? 232 00:09:51,610 --> 00:09:58,470 Once again, I like to personally separate out the 233 00:09:58,470 --> 00:10:00,029 number that goes into all of them. 234 00:10:00,029 --> 00:10:01,319 And all of these are divisible by 5. 235 00:10:01,320 --> 00:10:03,260 And that just simplifies it in my head. 236 00:10:03,259 --> 00:10:05,759 So if I divide all of these by 5-- actually, I could just 237 00:10:05,759 --> 00:10:08,600 divide both sides of this equation by 5. 238 00:10:08,600 --> 00:10:10,409 0 divided by 5 is 0. 239 00:10:10,409 --> 00:10:14,909 And then that left side divided by 5 becomes 2x 240 00:10:14,909 --> 00:10:27,959 squared minus 5x plus 3 is equal to 0. 241 00:10:27,960 --> 00:10:30,400 So if this is 2x squared here, so it's going to be two 242 00:10:30,399 --> 00:10:33,879 numbers when you multiply them equal 3 and when you-- so 243 00:10:33,879 --> 00:10:36,210 let's think about this a little bit. 244 00:10:36,210 --> 00:10:37,830 Actually, let me write it down here because I think I'll need 245 00:10:37,830 --> 00:10:38,930 more space. 246 00:10:38,929 --> 00:10:44,239 2x squared minus 5x plus 3 is equal to 0. 247 00:10:44,240 --> 00:10:46,039 And I just divided both sides of the equation 248 00:10:46,039 --> 00:10:47,754 by 5 to get to this. 249 00:10:47,754 --> 00:10:50,429 So let's see what we can do here. 250 00:10:50,429 --> 00:10:52,870 So we have a 2x squared here and they already kind of 251 00:10:52,870 --> 00:10:54,860 hinted to us that we're going to have an integer solution, 252 00:10:54,860 --> 00:10:56,230 so we can factor this. 253 00:10:56,230 --> 00:11:00,460 So the intuition is that this is going to be 2x times-- you 254 00:11:00,460 --> 00:11:02,250 know, plus something. 255 00:11:02,250 --> 00:11:03,460 Plus a. 256 00:11:03,460 --> 00:11:05,650 Times-- well what times? 257 00:11:05,649 --> 00:11:06,809 Times probably x, right? 258 00:11:06,809 --> 00:11:09,339 2x times x is 2x squared. 259 00:11:09,340 --> 00:11:11,050 Now that wouldn't be completely obvious if they 260 00:11:11,049 --> 00:11:12,969 didn't already tell us that we could factor this. 261 00:11:12,970 --> 00:11:13,550 You might have to use a 262 00:11:13,549 --> 00:11:14,839 quadratic equation or something. 263 00:11:14,840 --> 00:11:16,660 Actually, the quadratic equation wouldn't be something 264 00:11:16,659 --> 00:11:18,799 crazy to use here because you can just 265 00:11:18,799 --> 00:11:19,889 kind of plug and chug. 266 00:11:19,889 --> 00:11:21,210 But let's see if we can get the intuition. 267 00:11:21,210 --> 00:11:25,470 So it's going to be 2x plus something times x plus 268 00:11:25,470 --> 00:11:26,830 something else. 269 00:11:26,830 --> 00:11:30,310 If we were to multiply this out, you get 2x times x is 2x 270 00:11:30,309 --> 00:11:31,379 squared as it should. 271 00:11:31,379 --> 00:11:35,250 2x times b is plus 2bx. 272 00:11:35,250 --> 00:11:38,179 a times x is plus ax. 273 00:11:38,179 --> 00:11:42,319 a times b is plus ab. 274 00:11:42,320 --> 00:11:43,570 And so let's see what we get. 275 00:11:43,570 --> 00:11:47,770 276 00:11:47,769 --> 00:11:55,399 So plus 2b plus ax plus ab. 277 00:11:55,399 --> 00:11:57,419 2x squared. 278 00:11:57,419 --> 00:11:59,179 OK, now can do pattern matching. 279 00:11:59,179 --> 00:12:01,439 This was our original thing. 280 00:12:01,440 --> 00:12:05,890 So 2 times b plus a has to be equal to-- this term is the 281 00:12:05,889 --> 00:12:08,149 same thing as this term right here. 282 00:12:08,149 --> 00:12:10,079 And that term has to be the same thing as 283 00:12:10,080 --> 00:12:13,440 that term right there. 284 00:12:13,440 --> 00:12:17,230 So first of all, I have a positive 3 here. 285 00:12:17,230 --> 00:12:19,289 So I'm multiplying two numbers to get a positive 3. 286 00:12:19,289 --> 00:12:23,939 So they either have to be both positive or both negative. 287 00:12:23,940 --> 00:12:26,460 And then the other interesting thing is we have-- when I take 288 00:12:26,460 --> 00:12:28,070 2 times one of them plus the other one, I 289 00:12:28,070 --> 00:12:29,920 get a negative number. 290 00:12:29,919 --> 00:12:32,519 So the only way when you're dealing with negative numbers, 291 00:12:32,519 --> 00:12:34,750 and when you just multiply it times a positive and add them 292 00:12:34,750 --> 00:12:37,049 to each other you get another negative number, is if they're 293 00:12:37,049 --> 00:12:38,339 both negative. 294 00:12:38,340 --> 00:12:39,860 This told us they both have to be negative 295 00:12:39,860 --> 00:12:41,050 because this is positive. 296 00:12:41,049 --> 00:12:44,459 And then since when you add them without any negative 297 00:12:44,460 --> 00:12:46,509 signs you get a negative number, it tells you that that 298 00:12:46,509 --> 00:12:47,919 has to be negative as well. 299 00:12:47,919 --> 00:12:49,189 So let's see. 300 00:12:49,190 --> 00:12:54,390 Let's just try 3-- negative 3 and negative 1. 301 00:12:54,389 --> 00:12:56,299 If negative 3 and negative 1. 302 00:12:56,299 --> 00:12:56,799 So you're right. 303 00:12:56,799 --> 00:12:57,329 Yeah. 304 00:12:57,330 --> 00:13:05,450 If b is equal to minus 1 and a is equal to minus 3, then 2 305 00:13:05,450 --> 00:13:08,050 times minus 1 is minus 2. 306 00:13:08,049 --> 00:13:09,079 Minus 3. 307 00:13:09,080 --> 00:13:12,460 Right, so b is equal to minus 1 and a is equal to minus 3. 308 00:13:12,460 --> 00:13:14,740 This is a bit of an art form here. 309 00:13:14,740 --> 00:13:16,950 I mean there isn't like a plug and chug, very mechanical way 310 00:13:16,950 --> 00:13:17,680 of doing this. 311 00:13:17,679 --> 00:13:20,609 The quadratic equation is one, but this is the best way that, 312 00:13:20,610 --> 00:13:23,220 at least, I know how to do these without. 313 00:13:23,220 --> 00:13:24,980 So, we know what a and b is. 314 00:13:24,980 --> 00:13:28,460 So it's 2x-- a is minus 3. 315 00:13:28,460 --> 00:13:34,910 2x minus 3 times x plus b. b is minus 1. 316 00:13:34,909 --> 00:13:38,289 So that's the factorization. 317 00:13:38,289 --> 00:13:41,589 So 2x minus 3 times x minus 1, which one? 318 00:13:41,590 --> 00:13:43,180 They have this one right here. 319 00:13:43,179 --> 00:13:45,089 2x minus 3. 320 00:13:45,090 --> 00:13:46,070 And I'm all out of time. 321 00:13:46,070 --> 00:13:47,790 See you in the next video.