1 00:00:00,000 --> 00:00:00,460 2 00:00:00,460 --> 00:00:04,610 In this video, I'm going to expose you to what is maybe 3 00:00:04,610 --> 00:00:07,860 one of at least the top five most useful formulas in 4 00:00:07,860 --> 00:00:08,550 mathematics. 5 00:00:08,550 --> 00:00:11,460 And if you've seen many of my videos, you know that I'm not 6 00:00:11,460 --> 00:00:13,660 a big fan of memorizing things. 7 00:00:13,660 --> 00:00:17,399 But I will recommend you memorize it with the caveat 8 00:00:17,399 --> 00:00:20,775 that you also remember how to prove it, because I don't want 9 00:00:20,775 --> 00:00:22,940 you to just remember things and not know 10 00:00:22,940 --> 00:00:23,870 where they came from. 11 00:00:23,870 --> 00:00:25,970 But with that said, let me show you what I'm talking 12 00:00:25,969 --> 00:00:32,329 about: it's the quadratic formula. 13 00:00:32,329 --> 00:00:37,619 And as you might guess, it is to solve for the roots, or the 14 00:00:37,619 --> 00:00:40,899 zeroes of quadratic equations. 15 00:00:40,899 --> 00:00:43,449 So let's speak in very general terms and I'll 16 00:00:43,450 --> 00:00:44,850 show you some examples. 17 00:00:44,850 --> 00:00:48,359 So let's say I have an equation of the form ax 18 00:00:48,359 --> 00:00:54,070 squared plus bx plus c is equal to 0. 19 00:00:54,070 --> 00:00:55,619 You should recognize this. 20 00:00:55,619 --> 00:01:00,449 This is a quadratic equation where a, b and c are-- Well, a 21 00:01:00,450 --> 00:01:02,730 is the coefficient on the x squared term or the second 22 00:01:02,729 --> 00:01:05,539 degree term, b is the coefficient on the x term and 23 00:01:05,540 --> 00:01:08,270 then c, is, you could imagine, the coefficient on the x to 24 00:01:08,269 --> 00:01:10,549 the zero term, or it's the constant term. 25 00:01:10,549 --> 00:01:13,549 Now, given that you have a general quadratic equation 26 00:01:13,549 --> 00:01:16,700 like this, the quadratic formula tells us that the 27 00:01:16,700 --> 00:01:24,100 solutions to this equation are x is equal to negative b plus 28 00:01:24,099 --> 00:01:32,019 or minus the square root of b squared minus 4ac, all 29 00:01:32,019 --> 00:01:34,530 of that over 2a. 30 00:01:34,530 --> 00:01:38,489 And I know it seems crazy and convoluted and hard for you to 31 00:01:38,489 --> 00:01:41,159 memorize right now, but as you get a lot more practice you'll 32 00:01:41,159 --> 00:01:44,170 see that it actually is a pretty reasonable formula to 33 00:01:44,170 --> 00:01:46,299 stick in your brain someplace. 34 00:01:46,299 --> 00:01:48,569 And you might say, gee, this is a wacky formula, where did 35 00:01:48,569 --> 00:01:49,339 it come from? 36 00:01:49,340 --> 00:01:50,659 And in the next video I'm going to show you 37 00:01:50,659 --> 00:01:51,259 where it came from. 38 00:01:51,260 --> 00:01:54,920 But I want you to get used to using it first. But it really 39 00:01:54,920 --> 00:01:57,670 just came from completing the square on this 40 00:01:57,670 --> 00:01:58,750 equation right there. 41 00:01:58,750 --> 00:02:00,879 If you complete the square here, you're actually going to 42 00:02:00,879 --> 00:02:04,250 get this solution and that is the quadratic 43 00:02:04,250 --> 00:02:06,390 formula, right there. 44 00:02:06,390 --> 00:02:09,949 So let's apply it to some problems. Let's start off with 45 00:02:09,949 --> 00:02:12,680 something that we could have factored just to verify that 46 00:02:12,680 --> 00:02:14,120 it's giving us the same answer. 47 00:02:14,120 --> 00:02:19,659 So let's say we have x squared plus 4x minus 48 00:02:19,659 --> 00:02:22,889 21 is equal to 0. 49 00:02:22,889 --> 00:02:30,229 So in this situation-- let me do that in a different color 50 00:02:30,229 --> 00:02:32,369 --a is equal to 1, right? 51 00:02:32,370 --> 00:02:34,414 The coefficient on the x squared term is 1. 52 00:02:34,414 --> 00:02:37,789 53 00:02:37,789 --> 00:02:40,909 b is equal to 4, the coefficient on the x-term. 54 00:02:40,909 --> 00:02:44,240 And then c is equal to negative 21, 55 00:02:44,240 --> 00:02:45,469 the constant term. 56 00:02:45,469 --> 00:02:49,859 And let's just plug it in the formula, so what do we get? 57 00:02:49,860 --> 00:02:54,470 We get x, this tells us that x is going to be equal to 58 00:02:54,469 --> 00:02:56,199 negative b. 59 00:02:56,199 --> 00:03:02,345 Negative b is negative 4-- I put the negative sign in front 60 00:03:02,346 --> 00:03:05,659 of that --negative b plus or minus the 61 00:03:05,659 --> 00:03:08,770 square root of b squared. 62 00:03:08,770 --> 00:03:11,230 b squared is 16, right? 63 00:03:11,229 --> 00:03:20,169 4 squared is 16, minus 4 times a, which is 1, times c, which 64 00:03:20,169 --> 00:03:21,869 is negative 21. 65 00:03:21,870 --> 00:03:24,750 So we can put a 21 out there and that negative sign will 66 00:03:24,750 --> 00:03:28,789 cancel out just like that with that-- Since this is the first 67 00:03:28,789 --> 00:03:31,400 time we're doing it, let me not skip too many steps. 68 00:03:31,400 --> 00:03:34,800 So negative 21, just so you can see how it fit in, and 69 00:03:34,800 --> 00:03:39,330 then all of that over 2a. 70 00:03:39,330 --> 00:03:42,760 a is 1, so all of that over 2. 71 00:03:42,759 --> 00:03:45,459 So what does this simplify, or hopefully it simplifies? 72 00:03:45,460 --> 00:03:50,120 So we get x is equal to negative 4 plus or minus the 73 00:03:50,120 --> 00:03:53,110 square root of-- Let's see we have a negative times a 74 00:03:53,110 --> 00:03:55,890 negative, that's going to give us a positive. 75 00:03:55,889 --> 00:04:00,469 And we had 16 plus, let's see this is 6, 4 times 1 is 4 76 00:04:00,469 --> 00:04:03,159 times 21 is 84. 77 00:04:03,159 --> 00:04:06,379 16 plus 84 is 100. 78 00:04:06,379 --> 00:04:07,810 That's nice. 79 00:04:07,810 --> 00:04:09,180 That's a nice perfect square. 80 00:04:09,180 --> 00:04:13,379 All of that over 2, and so this is going to be equal to 81 00:04:13,379 --> 00:04:19,159 negative 4 plus or minus 10 over 2. 82 00:04:19,160 --> 00:04:23,060 We could just divide both of these terms by 2 right now. 83 00:04:23,060 --> 00:04:27,839 So this is equal to negative 4 divided by 2 is negative 2 84 00:04:27,839 --> 00:04:32,000 plus or minus 10 divided by 2 is 5. 85 00:04:32,000 --> 00:04:38,490 So that tells us that x could be equal to negative 2 plus 5, 86 00:04:38,490 --> 00:04:44,930 which is 3, or x could be equal to negative 2 minus 5, 87 00:04:44,930 --> 00:04:47,240 which is negative 7. 88 00:04:47,240 --> 00:04:49,420 So the quadratic formula seems to have given us 89 00:04:49,420 --> 00:04:50,500 an answer for this. 90 00:04:50,500 --> 00:04:53,240 You can verify just by substituting back in that 91 00:04:53,240 --> 00:04:55,840 these do work, or you could even just try to factor this 92 00:04:55,839 --> 00:04:56,609 right here. 93 00:04:56,610 --> 00:04:59,110 You say what two numbers when you take their product, you 94 00:04:59,110 --> 00:05:02,230 get negative 21 and when you take their sum you get 95 00:05:02,230 --> 00:05:04,200 positive 4? 96 00:05:04,199 --> 00:05:08,740 So you'd get x plus 7 times x minus 3 is 97 00:05:08,740 --> 00:05:10,689 equal to negative 21. 98 00:05:10,689 --> 00:05:14,969 Notice 7 times negative 3 is negative 21, 7 minus 3 is 99 00:05:14,970 --> 00:05:16,160 positive 4. 100 00:05:16,160 --> 00:05:20,850 You would get x plus-- sorry it's not negative --21 is 101 00:05:20,850 --> 00:05:22,590 equal to 0. 102 00:05:22,589 --> 00:05:23,889 There should be a 0 there. 103 00:05:23,889 --> 00:05:29,379 So you get x plus 7 is equal to 0, or x minus 104 00:05:29,379 --> 00:05:30,810 3 is equal to 0. 105 00:05:30,810 --> 00:05:36,850 X could be equal to negative 7 or x could be equal to 3. 106 00:05:36,850 --> 00:05:39,390 So it definitely gives us the same answer as factoring, so 107 00:05:39,389 --> 00:05:42,699 you might say, hey why bother with this crazy mess? 108 00:05:42,699 --> 00:05:45,139 And the reason we want to bother with this crazy mess is 109 00:05:45,139 --> 00:05:48,149 it'll also work for problems that are hard to factor. 110 00:05:48,149 --> 00:05:50,209 And let's do a couple of those, let's do some 111 00:05:50,209 --> 00:05:53,079 hard-to-factor problems right now. 112 00:05:53,079 --> 00:05:57,409 So let's scroll down to get some fresh real estate. 113 00:05:57,410 --> 00:06:01,150 Let's rewrite the formula again, just in case we haven't 114 00:06:01,149 --> 00:06:04,579 had it memorized yet. x is going to be equal to negative 115 00:06:04,579 --> 00:06:11,180 b plus or minus the square root of b squared minus 4ac, 116 00:06:11,180 --> 00:06:14,480 all of that over 2a. 117 00:06:14,480 --> 00:06:17,540 I'll supply this to another problem. 118 00:06:17,540 --> 00:06:30,319 Let's say we have the equation 3x squared plus 6x is equal to 119 00:06:30,319 --> 00:06:31,719 negative 10. 120 00:06:31,720 --> 00:06:33,610 Well, the first thing we want to do is get it in the form 121 00:06:33,610 --> 00:06:35,980 where all of our terms or on the left-hand side, so let's 122 00:06:35,980 --> 00:06:38,450 add 10 to both sides of this equation. 123 00:06:38,449 --> 00:06:46,680 We get 3x squared plus the 6x plus 10 is equal to 0. 124 00:06:46,680 --> 00:06:49,930 And now we can use a quadratic formula. 125 00:06:49,930 --> 00:06:51,319 So let's apply it here. 126 00:06:51,319 --> 00:06:53,050 So a is equal to 3. 127 00:06:53,050 --> 00:06:59,900 That is a, this is b and this right here is c. 128 00:06:59,899 --> 00:07:01,870 So the quadratic formula tells us the 129 00:07:01,870 --> 00:07:04,060 solutions to this equation. 130 00:07:04,060 --> 00:07:09,569 The roots of this quadratic function, I guess 131 00:07:09,569 --> 00:07:10,569 we could call it. 132 00:07:10,569 --> 00:07:13,300 x is going to be equal to negative b. 133 00:07:13,300 --> 00:07:17,920 b is 6, so negative 6 plus or minus the 134 00:07:17,920 --> 00:07:21,350 square root of b squared. 135 00:07:21,350 --> 00:07:29,270 b is 6, so we get 6 squared minus 4 times a, which is 3 136 00:07:29,269 --> 00:07:32,629 times c, which is 10. 137 00:07:32,629 --> 00:07:37,050 Let's stretch out the radical little bit, all of that over 2 138 00:07:37,050 --> 00:07:42,189 times a, 2 times 3. 139 00:07:42,189 --> 00:07:46,379 So we get x is equal to negative 6 plus or minus the 140 00:07:46,379 --> 00:07:57,319 square root of 36 minus-- this is interesting --minus 4 times 141 00:07:57,319 --> 00:07:59,089 3 times 10. 142 00:07:59,089 --> 00:08:05,689 So this is minus-- 4 times 3 times 10. 143 00:08:05,689 --> 00:08:11,500 So this is minus 120. 144 00:08:11,500 --> 00:08:15,009 All of that over 6. 145 00:08:15,009 --> 00:08:17,750 So this is interesting, you might already realize why it's 146 00:08:17,750 --> 00:08:18,220 interesting. 147 00:08:18,220 --> 00:08:19,970 What is this going to simplify to? 148 00:08:19,970 --> 00:08:23,610 36 minus 120 is what? 149 00:08:23,610 --> 00:08:30,629 That's 84. 150 00:08:30,629 --> 00:08:33,860 We make this into a 10, this will become an 151 00:08:33,860 --> 00:08:36,158 11, this is a 4. 152 00:08:36,158 --> 00:08:39,590 It is 84, so this is going to be equal to negative 6 plus or 153 00:08:39,590 --> 00:08:42,798 minus the square root of-- But not positive 84, that's if 154 00:08:42,798 --> 00:08:44,639 it's 120 minus 36. 155 00:08:44,639 --> 00:08:46,919 We have 36 minus 120. 156 00:08:46,919 --> 00:08:53,629 It's going to be negative 84 all of that 6. 157 00:08:53,629 --> 00:08:55,139 So you might say, gee, this is crazy. 158 00:08:55,139 --> 00:08:56,929 What a this silly quadratic formula you're 159 00:08:56,929 --> 00:08:58,000 introducing me to, Sal? 160 00:08:58,000 --> 00:08:58,789 It's worthless. 161 00:08:58,789 --> 00:09:03,019 It just gives me a square root of a negative number. 162 00:09:03,019 --> 00:09:04,379 It's not giving me an answer. 163 00:09:04,379 --> 00:09:06,750 And the reason why it's not giving you an answer, at least 164 00:09:06,750 --> 00:09:11,450 an answer that you might want, is because this will have no 165 00:09:11,450 --> 00:09:12,700 real solutions. 166 00:09:12,700 --> 00:09:17,910 167 00:09:17,909 --> 00:09:20,259 In the future, we're going to introduce something called an 168 00:09:20,259 --> 00:09:23,309 imaginary number, which is a square root of a negative 169 00:09:23,309 --> 00:09:27,719 number, and then we can actually express this in terms 170 00:09:27,720 --> 00:09:28,810 of those numbers. 171 00:09:28,809 --> 00:09:31,539 So this actually does have solutions, but they involve 172 00:09:31,539 --> 00:09:32,659 imaginary numbers. 173 00:09:32,659 --> 00:09:34,689 So this actually has no real solutions, we're taking the 174 00:09:34,690 --> 00:09:37,760 square root of a negative number. 175 00:09:37,759 --> 00:09:41,279 So the b squared with the b squared minus 4ac, if this 176 00:09:41,279 --> 00:09:45,329 term right here is negative, then you're not going to have 177 00:09:45,330 --> 00:09:47,290 any real solutions. 178 00:09:47,289 --> 00:09:48,610 And let's verify that for ourselves. 179 00:09:48,610 --> 00:09:51,240 Let's get our graphic calculator out and let's graph 180 00:09:51,240 --> 00:09:54,600 this equation right here. 181 00:09:54,600 --> 00:09:57,710 So, let's get the graphs that y is equal to-- that's what I 182 00:09:57,710 --> 00:10:10,509 had there before --3x squared plus 6x plus 10. 183 00:10:10,509 --> 00:10:11,919 So that's the equation and we're going to see where it 184 00:10:11,919 --> 00:10:13,209 intersects the x-axis. 185 00:10:13,210 --> 00:10:15,370 Where does it equal 0? 186 00:10:15,370 --> 00:10:17,149 So let me graph it. 187 00:10:17,149 --> 00:10:20,509 188 00:10:20,509 --> 00:10:23,700 Notice, this thing just comes down and then goes back up. 189 00:10:23,700 --> 00:10:26,879 Its vertex is sitting here above the x-axis and it's 190 00:10:26,879 --> 00:10:27,710 upward-opening. 191 00:10:27,710 --> 00:10:30,290 It never intersects the x-axis. 192 00:10:30,289 --> 00:10:33,261 So at no point will this expression, will this 193 00:10:33,261 --> 00:10:34,649 function, equal 0. 194 00:10:34,649 --> 00:10:38,529 At no point will y equal 0 on this graph. 195 00:10:38,529 --> 00:10:41,699 So once again, the quadratic formula seems to be working. 196 00:10:41,700 --> 00:10:44,710 Let's do one more example, you can ever see 197 00:10:44,710 --> 00:10:46,730 enough examples here. 198 00:10:46,730 --> 00:10:50,300 And I want to do ones that are, you know, maybe not so 199 00:10:50,299 --> 00:10:52,159 obvious to factor. 200 00:10:52,159 --> 00:11:01,929 So let's say we get negative 3x squared plus 12x plus 1 is 201 00:11:01,929 --> 00:11:03,769 equal to 0. 202 00:11:03,769 --> 00:11:06,449 Now let's try to do it just having the quadratic formula 203 00:11:06,450 --> 00:11:07,740 in our brain. 204 00:11:07,740 --> 00:11:12,289 So the x's that satisfy this equation are going to be 205 00:11:12,289 --> 00:11:13,689 negative b. 206 00:11:13,690 --> 00:11:18,580 This is b So negative b is negative 12 plus or minus the 207 00:11:18,580 --> 00:11:27,300 square root of b squared, of 144, that's b squared minus 4 208 00:11:27,299 --> 00:11:33,870 times a, which is negative 3 times c, which is 1, all of 209 00:11:33,870 --> 00:11:39,539 that over 2 times a, over 2 times negative 3. 210 00:11:39,539 --> 00:11:43,969 So all of that over negative 6, this is going to be equal 211 00:11:43,970 --> 00:11:47,629 to negative 12 plus or minus the square root 212 00:11:47,629 --> 00:11:49,029 of-- What is this? 213 00:11:49,029 --> 00:11:51,709 It's a negative times a negative so they cancel out. 214 00:11:51,710 --> 00:12:01,220 So I have 144 plus 12, so that is 156, right? 215 00:12:01,220 --> 00:12:06,910 144 plus 12, all of that over negative 6. 216 00:12:06,909 --> 00:12:10,350 Now, I suspect we can simplify this 156. 217 00:12:10,350 --> 00:12:12,269 We could maybe bring some things out 218 00:12:12,269 --> 00:12:13,850 of the radical sign. 219 00:12:13,850 --> 00:12:17,170 So let's attempt to do that. 220 00:12:17,169 --> 00:12:20,019 So let's do a prime factorization of 156. 221 00:12:20,019 --> 00:12:23,600 Sometimes, this is the hardest part, simplifying the radical. 222 00:12:23,600 --> 00:12:30,080 So 156 is the same thing as 2 times 78. 223 00:12:30,080 --> 00:12:33,950 78 is the same thing as 2 times what? 224 00:12:33,950 --> 00:12:39,800 That's 2 times 39. 225 00:12:39,799 --> 00:12:48,959 So the square root of 156 is equal to the square root of 2 226 00:12:48,960 --> 00:12:52,800 times 2 times 39 or we could say that's the square root of 227 00:12:52,799 --> 00:12:57,109 2 times 2 times the square root of 39. 228 00:12:57,110 --> 00:12:58,810 And this, obviously, is just going to be the square root of 229 00:12:58,809 --> 00:13:01,829 4 or this is the square root of 2 times 2 is just 2. 230 00:13:01,830 --> 00:13:06,310 2 square roots of 39, if I did that properly, let's 231 00:13:06,309 --> 00:13:07,709 see, 4 times 39. 232 00:13:07,710 --> 00:13:10,150 Yeah, it looks like it's right. 233 00:13:10,149 --> 00:13:17,590 So this up here will simplify to negative 12 plus or minus 2 234 00:13:17,590 --> 00:13:23,220 times the square root of 39, all of that over negative 6. 235 00:13:23,220 --> 00:13:25,649 Now we can divide the numerator and the denominator 236 00:13:25,649 --> 00:13:26,590 maybe by 2. 237 00:13:26,590 --> 00:13:32,060 So this will be equal to negative 6 plus or minus the 238 00:13:32,059 --> 00:13:36,019 square root of 39 over negative 3. 239 00:13:36,019 --> 00:13:38,539 Or we could separate these two terms out. 240 00:13:38,539 --> 00:13:42,799 We could say this is equal to negative 6 over negative 3 241 00:13:42,799 --> 00:13:49,309 plus or minus the square root of 39 over negative 3. 242 00:13:49,309 --> 00:13:51,399 Now, this is just a 2 right here, right? 243 00:13:51,399 --> 00:13:55,189 These cancel out, 6 divided by 3 is 2, so we get 2. 244 00:13:55,190 --> 00:13:57,810 And now notice, if this is plus and we use this minus 245 00:13:57,809 --> 00:13:59,779 sign, the plus will become negative and the negative will 246 00:13:59,779 --> 00:14:01,159 become positive. 247 00:14:01,159 --> 00:14:02,389 But it still doesn't matter, right? 248 00:14:02,389 --> 00:14:05,259 We could say minus or plus, that's the same thing as plus 249 00:14:05,259 --> 00:14:10,009 or minus the square root of 39 nine over 3. 250 00:14:10,009 --> 00:14:12,939 I think that's about as simple as we can get this answered. 251 00:14:12,940 --> 00:14:14,620 I want to make a very clear point of what I 252 00:14:14,620 --> 00:14:16,419 did that last step. 253 00:14:16,419 --> 00:14:19,529 I did not forget about this negative sign. 254 00:14:19,529 --> 00:14:20,429 I just said it doesn't matter. 255 00:14:20,429 --> 00:14:22,189 It's going to turn the positive into the negative; 256 00:14:22,190 --> 00:14:23,800 it's going to turn the negative into the positive. 257 00:14:23,799 --> 00:14:25,139 Let me rewrite this. 258 00:14:25,139 --> 00:14:29,659 So this right here can be rewritten as 2 plus the square 259 00:14:29,659 --> 00:14:36,209 root of 39 over negative 3 or 2 minus the square root of 39 260 00:14:36,210 --> 00:14:38,080 over negative 3, right? 261 00:14:38,080 --> 00:14:40,610 That's what the plus or minus means, it could be this or 262 00:14:40,610 --> 00:14:42,600 that or both of them, really. 263 00:14:42,600 --> 00:14:45,430 Now in this situation, this negative 3 will turn into 2 264 00:14:45,429 --> 00:14:49,289 minus the square root of 39 over 3, right? 265 00:14:49,289 --> 00:14:51,149 I'm just taking this negative out. 266 00:14:51,149 --> 00:14:53,129 Here the negative and the negative will become a 267 00:14:53,129 --> 00:14:56,090 positive, and you get 2 plus the square root 268 00:14:56,090 --> 00:14:59,210 of 39 over 3, right? 269 00:14:59,210 --> 00:15:01,129 A negative times a negative is a positive. 270 00:15:01,129 --> 00:15:04,399 So once again, you have 2 plus or minus the 271 00:15:04,399 --> 00:15:07,189 square of 39 over 3. 272 00:15:07,190 --> 00:15:11,550 2 plus or minus the square root of 39 over 3 are 273 00:15:11,549 --> 00:15:16,794 solutions to this equation right there. 274 00:15:16,794 --> 00:15:17,299 Let verify. 275 00:15:17,299 --> 00:15:19,919 I'm just curious what the graph looks like. 276 00:15:19,919 --> 00:15:24,360 So let's just look at it. 277 00:15:24,360 --> 00:15:25,860 Let me clear this. 278 00:15:25,860 --> 00:15:27,669 Where is the clear button? 279 00:15:27,669 --> 00:15:37,279 So we have negative 3 three squared plus 12x plus 1 and 280 00:15:37,279 --> 00:15:39,721 let's graph it. 281 00:15:39,721 --> 00:15:42,279 Let's see where it intersects the x-axis. 282 00:15:42,279 --> 00:15:47,179 It goes up there and then back down again. 283 00:15:47,179 --> 00:15:51,839 So 2 plus or minus the square, you see-- The square root of 284 00:15:51,840 --> 00:15:56,330 39 is going to be a little bit more than 6, right? 285 00:15:56,330 --> 00:15:58,090 Because 36 is 6 squared. 286 00:15:58,090 --> 00:16:00,910 So it's going be a little bit more than 6, so this is going 287 00:16:00,909 --> 00:16:02,110 to be a little bit more than 2. 288 00:16:02,110 --> 00:16:03,889 A little bit more than 6 divided by 2 is a little bit 289 00:16:03,889 --> 00:16:04,615 more than 2. 290 00:16:04,615 --> 00:16:06,580 So you're going to get one value that's a little bit more 291 00:16:06,580 --> 00:16:09,100 than 4 and then another value that should be a little bit 292 00:16:09,100 --> 00:16:10,690 less than 1. 293 00:16:10,690 --> 00:16:13,590 And that looks like the case, you have 1, 2, 3, 4. 294 00:16:13,590 --> 00:16:18,460 You have a value that's pretty close to 4, and then you have 295 00:16:18,460 --> 00:16:25,019 another value that is a little bit-- It looks close to 0 but 296 00:16:25,019 --> 00:16:26,539 maybe a little bit less than that. 297 00:16:26,539 --> 00:16:29,089 So anyway, hopefully you found this application of the 298 00:16:29,090 --> 00:16:31,259 quadratic formula helpful. 299 00:16:31,259 --> 00:16:31,732