1 00:00:05,220 --> 00:00:07,600 What are we getting an equation here? 2 00:00:13,220 --> 00:00:15,660 direction or solve with me. Solve 3 00:00:15,660 --> 00:00:17,740 find the solution set. 4 00:00:17,740 --> 00:00:18,380 That's great. 5 00:00:19,280 --> 00:00:20,520 Set of all solutions. 6 00:00:22,360 --> 00:00:24,580 What do we call this type of equation? 7 00:00:29,440 --> 00:00:32,100 This is called a linear equation. 8 00:00:32,100 --> 00:00:35,280 Why is this called a linear equation? 9 00:00:35,440 --> 00:00:38,940 The variable is only to the first power. 10 00:00:41,920 --> 00:00:42,680 I'll just tell you. 11 00:00:43,500 --> 00:00:44,840 This is what type of polynomial? 12 00:00:45,800 --> 00:00:46,740 Linear polynomial. 13 00:00:48,080 --> 00:00:50,880 A linear polynomial. Yes, it happens to be binomial. 14 00:00:52,200 --> 00:00:55,880 So this is a linear polynomial. 15 00:00:57,080 --> 00:00:57,860 And what is this? 16 00:01:00,000 --> 00:01:00,080 Constant. 17 00:01:02,820 --> 00:01:03,540 Polynomial. 18 00:01:04,000 --> 00:01:05,180 Definitely a monomial. 19 00:01:06,260 --> 00:01:06,280 Okay. 20 00:01:09,120 --> 00:01:10,840 The linear is the larger degree. 21 00:01:12,560 --> 00:01:14,220 Since it's the larger degree, 22 00:01:15,060 --> 00:01:18,100 the entire thing is called linear. Sorry, a linear equation. 23 00:01:19,120 --> 00:01:20,860 A linear equation. 24 00:01:23,320 --> 00:01:25,820 You could call it a linear polynomial equation. 25 00:01:26,760 --> 00:01:28,320 We simplify it. Linear equation. 26 00:01:29,680 --> 00:01:32,300 And we do a bunch of steps. 27 00:01:32,960 --> 00:01:33,780 What's the first step? 28 00:01:34,420 --> 00:01:36,720 Subtract seven. 29 00:01:38,660 --> 00:01:40,280 We get 3x 30 00:01:42,320 --> 00:01:43,440 equals two. 31 00:01:43,960 --> 00:01:46,680 And then we divide by three. 32 00:01:47,140 --> 00:01:47,800 Divide by three. 33 00:01:49,200 --> 00:01:50,520 Just two thirds. 34 00:01:51,280 --> 00:01:53,600 And then the solution is 35 00:01:54,800 --> 00:01:56,780 two thirds. 36 00:01:57,780 --> 00:01:59,580 And the solution set is 37 00:02:00,680 --> 00:02:02,720 the set containing two thirds. 38 00:02:05,000 --> 00:02:06,880 This process we went through, 39 00:02:07,360 --> 00:02:09,220 we learned in, I don't know, seventh grade? 40 00:02:09,580 --> 00:02:13,260 Is often known as isolating the variable. 41 00:02:23,820 --> 00:02:26,340 Once we got the variable all by itself, 42 00:02:26,700 --> 00:02:28,580 it's obvious what the solution is. 43 00:02:29,020 --> 00:02:30,340 It's the number on the other side. 44 00:02:31,420 --> 00:02:33,700 Because when you replace it for the x, 45 00:02:34,680 --> 00:02:35,900 makes the equation true. 46 00:02:37,760 --> 00:02:38,180 Okay. 47 00:02:41,660 --> 00:02:42,080 Next. 48 00:03:03,720 --> 00:03:05,240 What do we call this side of the equation? 49 00:03:07,280 --> 00:03:08,960 It's a quadratic equation. 50 00:03:10,100 --> 00:03:12,320 The left side is a quadratic polynomial, 51 00:03:12,400 --> 00:03:13,940 the right side is a constant polynomial, 52 00:03:14,300 --> 00:03:16,260 the larger the two degrees is two, 53 00:03:16,380 --> 00:03:17,240 the quadratic. 54 00:03:17,580 --> 00:03:20,760 We call the entire thing a quadratic equation. 55 00:03:23,480 --> 00:03:26,540 This is what we're talking about solving quadratic equations today. 56 00:03:31,120 --> 00:03:32,680 So, what are we going to do? 57 00:03:47,820 --> 00:03:50,280 We're going to isolate the variable. 58 00:03:51,800 --> 00:03:59,520 We want to get x equals how we got a problem. 59 00:03:59,760 --> 00:04:00,320 What's the problem? 60 00:04:04,280 --> 00:04:05,880 There's an x squared and there's an x. 61 00:04:06,420 --> 00:04:08,560 There's no way to combine those into one, right? 62 00:04:09,160 --> 00:04:10,540 Because they're different terms. 63 00:04:11,760 --> 00:04:12,140 You got an idea? 64 00:04:14,720 --> 00:04:16,440 Okay, we could factor out the x. 65 00:04:18,720 --> 00:04:20,120 Let's do that for a second. 66 00:04:23,020 --> 00:04:23,940 The x plus two. 67 00:04:35,880 --> 00:04:37,480 But there's no way to still... 68 00:04:37,480 --> 00:04:39,880 We still have two different x's. 69 00:04:44,100 --> 00:04:44,560 So now what? 70 00:04:45,880 --> 00:04:49,080 Subtract 15 and make it a quadratic trinomial. 71 00:04:49,420 --> 00:04:50,440 We want to factor them out. 72 00:04:51,980 --> 00:04:53,560 Okay, why would we do that? 73 00:04:54,040 --> 00:04:55,580 Because it will equal zero. 74 00:05:04,160 --> 00:05:06,260 Okay, how about this? 75 00:05:09,820 --> 00:05:13,940 Well, we're looking for a number and a number plus two 76 00:05:13,940 --> 00:05:15,340 more than that same number. 77 00:05:16,040 --> 00:05:17,500 It equals 15. 78 00:05:21,240 --> 00:05:23,440 We could start guessing and checking, 79 00:05:23,600 --> 00:05:24,520 but that's a terrible way. 80 00:05:25,820 --> 00:05:28,800 Okay, we have a very, very important property. 81 00:05:30,900 --> 00:05:33,500 Bet you even know what it is. 82 00:05:35,660 --> 00:05:38,540 It's called the zero product property. 83 00:05:38,540 --> 00:05:38,580 Okay. 84 00:05:48,800 --> 00:05:54,540 If a times b equals zero... 85 00:05:57,340 --> 00:06:01,540 If two numbers multiplied to be zero... 86 00:06:02,340 --> 00:06:03,600 What has to be true? 87 00:06:06,300 --> 00:06:07,600 Either a is zero... 88 00:06:11,860 --> 00:06:13,260 Or b equals zero. 89 00:06:14,840 --> 00:06:16,800 Or they both could be zero, correct? 90 00:06:18,180 --> 00:06:20,900 Okay, so that's still the zero product property. 91 00:06:25,620 --> 00:06:26,820 Why is it called that? 92 00:06:27,100 --> 00:06:29,340 Then why is it called the zero product property? 93 00:06:30,260 --> 00:06:31,000 Because it equals zero. 94 00:06:31,520 --> 00:06:33,840 Yeah, because we have a product that equals zero. 95 00:06:35,420 --> 00:06:36,900 This product equals zero. 96 00:06:37,760 --> 00:06:39,240 It's called the zero product property. 97 00:06:39,340 --> 00:06:43,180 Sometimes it's called the zero factor property. 98 00:06:44,540 --> 00:06:48,440 We have two factors that equal zero, 99 00:06:48,560 --> 00:06:50,560 but we're just going to call it the zero product property. 100 00:06:52,800 --> 00:06:53,700 Okay, so let's go back here. 101 00:06:55,420 --> 00:06:59,140 We have a product that equals 15. 102 00:07:02,500 --> 00:07:03,060 Okay. 103 00:07:08,380 --> 00:07:09,500 There's no... 104 00:07:09,500 --> 00:07:11,220 Well, let me give you this. 105 00:07:11,320 --> 00:07:13,720 What are some numbers that multiply to 15? 106 00:07:15,100 --> 00:07:16,500 Three and five. 107 00:07:17,340 --> 00:07:17,520 What else? 108 00:07:18,340 --> 00:07:18,740 15 and one. 109 00:07:20,080 --> 00:07:21,780 15 and one. What else? 110 00:07:23,660 --> 00:07:24,780 Negative three, negative five. 111 00:07:24,820 --> 00:07:26,940 Negative three, negative five. 112 00:07:27,260 --> 00:07:27,600 What else? 113 00:07:27,760 --> 00:07:29,040 Negative three, negative five. 114 00:07:30,220 --> 00:07:31,820 What else? 115 00:07:35,720 --> 00:07:38,260 How about 30 and one half? 116 00:07:40,400 --> 00:07:41,320 Negative 30, negative one half. 117 00:07:41,760 --> 00:07:44,500 How about 60 and a fourth? 118 00:07:45,840 --> 00:07:50,380 How many numbers multiply to be 15? 119 00:07:51,440 --> 00:07:52,740 More than a lot. 120 00:07:54,200 --> 00:07:54,600 An infinite. 121 00:07:54,600 --> 00:07:55,520 Okay. 122 00:07:56,820 --> 00:08:02,460 So we don't have a 15 product property, 123 00:08:04,200 --> 00:08:06,400 but we do have a zero product property. 124 00:08:07,380 --> 00:08:10,320 If we have zero, we've got only two options. 125 00:08:10,720 --> 00:08:12,900 Either the first number is zero or the second number is zero. 126 00:08:16,360 --> 00:08:18,640 So we are going to start with this. 127 00:08:19,860 --> 00:08:20,620 So we're going to go backwards. 128 00:08:21,300 --> 00:08:22,840 We have to have zero on one side. 129 00:08:22,840 --> 00:08:26,900 So we start by subtracting 15. 130 00:08:28,660 --> 00:08:30,240 Yes, people did say this before. 131 00:08:33,400 --> 00:08:36,740 And now we need to change this to a product. 132 00:08:40,100 --> 00:08:41,840 And that process is called... 133 00:08:44,040 --> 00:08:44,140 Factoring. 134 00:08:44,140 --> 00:08:44,580 Factoring. 135 00:08:44,820 --> 00:08:45,100 Thank you. 136 00:08:46,000 --> 00:08:49,620 We want to change our sum to a product that's called factoring, 137 00:08:50,260 --> 00:08:51,120 to write as a product. 138 00:08:51,120 --> 00:08:53,420 And so we're going to factor this. 139 00:08:56,820 --> 00:08:57,640 What's this factor to? 140 00:09:01,320 --> 00:09:03,240 Five and negative three. 141 00:09:03,480 --> 00:09:04,540 Five and negative three. 142 00:09:04,900 --> 00:09:05,340 Oops. 143 00:09:09,320 --> 00:09:09,940 Okay. 144 00:09:12,700 --> 00:09:15,860 Now we have a product that equals zero. 145 00:09:15,860 --> 00:09:17,980 Oh, so it's... 146 00:09:17,980 --> 00:09:25,380 By the zero product property, 147 00:09:28,860 --> 00:09:32,980 this number, x minus three, equals zero. 148 00:09:33,660 --> 00:09:38,960 Or this number, x plus five, equals zero. 149 00:09:42,080 --> 00:09:45,880 Now we don't have a quadratic equation. 150 00:09:46,180 --> 00:09:47,240 We have two what's that? 151 00:09:48,040 --> 00:09:48,920 Linear. 152 00:09:49,040 --> 00:09:50,180 Linear equations. 153 00:09:51,440 --> 00:09:55,780 And now all we have to do is isolate the variable. 154 00:09:59,480 --> 00:10:01,240 Add three. 155 00:10:04,080 --> 00:10:07,120 And the solution is three. 156 00:10:08,120 --> 00:10:10,800 And for x plus five, subtract five from both sides. 157 00:10:11,660 --> 00:10:13,780 Solution is negative five. 158 00:10:15,400 --> 00:10:15,420 Okay. 159 00:10:17,360 --> 00:10:19,200 And there it is. 160 00:10:20,120 --> 00:10:23,060 So once we turned it into two linear equations, 161 00:10:23,060 --> 00:10:24,780 we're able to just isolate the variable 162 00:10:26,260 --> 00:10:28,200 and get our solution set. 163 00:10:28,440 --> 00:10:31,540 And our solution set is negative five and three. 164 00:10:33,900 --> 00:10:34,580 Okay. 165 00:10:35,820 --> 00:10:38,620 So this is how we're going to solve quadratic equations. 166 00:10:40,320 --> 00:10:42,200 We need one side equals zero. 167 00:10:43,360 --> 00:10:44,700 Then we're going to factor. 168 00:10:46,540 --> 00:10:46,840 Okay. 169 00:10:47,920 --> 00:10:50,360 And then we use the zero product property, 170 00:10:50,420 --> 00:10:52,600 write two separate linear equations. 171 00:10:54,540 --> 00:10:58,500 And then we get our two solutions, if there are two solutions. 172 00:11:00,500 --> 00:11:01,000 Is that cool? 173 00:11:02,060 --> 00:11:02,340 Okay. 174 00:11:02,340 --> 00:11:04,080 We solved our first quadratic equation. 175 00:11:06,340 --> 00:11:09,260 This was the major point of why we spent so much time factoring. 176 00:11:10,640 --> 00:11:11,040 Okay. 177 00:11:11,460 --> 00:11:13,660 It allows us to solve quadratic equations, 178 00:11:13,900 --> 00:11:15,400 which before was impossible. 179 00:11:16,220 --> 00:11:17,100 You didn't know how to factor, 180 00:11:17,360 --> 00:11:19,980 and we didn't even know what factoring was before. 181 00:11:21,020 --> 00:11:22,320 Let's try another one. 182 00:11:42,260 --> 00:11:43,660 Okay. 183 00:12:04,880 --> 00:12:06,580 So what's the first step? 184 00:12:09,820 --> 00:12:11,260 We must have zero on one side. 185 00:12:11,440 --> 00:12:12,100 We don't care what side. 186 00:12:13,200 --> 00:12:14,360 We must have zero on one side. 187 00:12:15,020 --> 00:12:16,520 So we subtract seven X. 188 00:12:18,820 --> 00:12:20,080 And what order are we in writing? 189 00:12:22,460 --> 00:12:23,180 Descending order. 190 00:12:30,600 --> 00:12:32,100 And once we do that, what do we do? 191 00:12:33,000 --> 00:12:33,980 We need a factor. 192 00:12:40,560 --> 00:12:41,680 And what's this factor, two? 193 00:12:44,120 --> 00:12:46,700 Two numbers that multiply to be negative 30, 194 00:12:46,880 --> 00:12:47,840 that add to be negative seven. 195 00:12:49,760 --> 00:12:51,960 Negative 10, positive three. 196 00:12:54,140 --> 00:12:54,160 Okay. 197 00:12:54,400 --> 00:12:58,180 And then once we have a fact or a product that equals zero, 198 00:12:58,260 --> 00:12:58,880 what do we use? 199 00:13:00,780 --> 00:13:02,000 Zero product. 200 00:13:03,800 --> 00:13:04,880 Zero product. 201 00:13:10,480 --> 00:13:12,320 Either X minus 10. 202 00:13:14,280 --> 00:13:16,780 This number equals zero. 203 00:13:17,800 --> 00:13:19,240 Or X plus three. 204 00:13:21,480 --> 00:13:23,780 And now we got two linear equations. 205 00:13:24,780 --> 00:13:27,780 And we solve those. 206 00:13:47,120 --> 00:13:50,860 And we got our two solutions. 207 00:13:54,980 --> 00:13:55,440 Okay. 208 00:13:57,160 --> 00:14:00,980 Quadratics can have at most two equations, 209 00:14:01,340 --> 00:14:02,320 sorry, two solutions. 210 00:14:03,620 --> 00:14:05,240 It could have one solution. 211 00:14:05,860 --> 00:14:07,180 We'll give an example of that in a minute. 212 00:14:08,760 --> 00:14:10,340 Or it can have no solutions. 213 00:14:11,060 --> 00:14:12,320 And we'll give an example of that. 214 00:14:12,620 --> 00:14:14,360 But most of the time it's going to have two solutions. 215 00:14:16,640 --> 00:14:17,760 Let's try another one. 216 00:14:39,260 --> 00:14:40,220 Step one. 217 00:14:42,640 --> 00:14:44,080 We're not setting it to zero. 218 00:14:44,600 --> 00:14:46,620 We need to get one side to be zero. 219 00:14:48,200 --> 00:14:49,460 And we do that by? 220 00:14:50,400 --> 00:14:51,420 Subtracting 100. 221 00:14:58,460 --> 00:15:01,200 And now we need to factor that quadratic. 222 00:15:07,480 --> 00:15:11,700 Factors of negative 300 that add to be negative 40. 223 00:15:29,100 --> 00:15:30,540 You know what, I screwed up. 224 00:15:32,720 --> 00:15:34,560 Let's make this a negative 100. 225 00:15:34,860 --> 00:15:37,300 Sorry, I messed up the problem. 226 00:15:39,300 --> 00:15:40,660 So this becomes positive. 227 00:15:43,580 --> 00:15:45,020 So sorry, put that in there. 228 00:15:46,340 --> 00:15:48,200 Otherwise it would not be able to be factored. 229 00:15:49,200 --> 00:15:58,452 Now we're looking for factors of 300 that add to be negative 40: negative 10 and 230 00:15:58,452 --> 00:15:59,840 negative 30. 231 00:16:02,760 --> 00:16:05,100 Negative 10, thank you. 232 00:16:05,940 --> 00:16:06,560 Negative 30. 233 00:16:09,420 --> 00:16:14,160 Minus 10x minus 30x plus 100. 234 00:16:18,200 --> 00:16:19,940 And factor it by grouping. 235 00:16:50,360 --> 00:16:58,360 Okay, once we factor, we have a factor that equals, or a product that equals zero. 236 00:16:59,380 --> 00:16:59,780 What's next? 237 00:17:03,160 --> 00:17:10,040 So by the zero product property. 238 00:17:12,020 --> 00:17:17,514 And when you're solving this for homework, I want you to write that in every time 239 00:17:17,514 --> 00:17:18,260 you use it. 240 00:17:19,580 --> 00:17:21,200 You can abbreviate it if you want. 241 00:17:21,840 --> 00:17:23,180 What should we abbreviate it as? 242 00:17:24,300 --> 00:17:24,620 ZPP. 243 00:17:26,580 --> 00:17:34,120 Either x minus 10 equals zero, or 3x minus 10 equals zero. 244 00:17:44,940 --> 00:17:48,960 And we have our two solutions, 10 thirds and 10. 245 00:17:50,720 --> 00:17:54,160 How can we check to see if our solutions are actual solutions? 246 00:17:56,740 --> 00:17:59,080 We can plug it back into the original and? 247 00:18:02,600 --> 00:18:03,980 What does it mean to be a solution? 248 00:18:06,860 --> 00:18:09,940 A number that, when replaced for the variable, makes the equation true. 249 00:18:10,360 --> 00:18:12,420 If you plug it in, it makes the equation true. 250 00:18:12,940 --> 00:18:15,580 If you factor right, it's going to be the solutions. 251 00:18:17,880 --> 00:18:18,320 Okay. 252 00:18:35,720 --> 00:18:36,580 This is a binomial. 253 00:18:37,720 --> 00:18:38,000 What do I write? 254 00:18:38,260 --> 00:18:38,420 Binomial. 255 00:18:40,000 --> 00:18:42,020 The only one we know how to factor is what? 256 00:18:42,020 --> 00:18:45,040 Difference of squares. 257 00:18:46,160 --> 00:18:47,080 Is this the difference of squares? 258 00:18:50,700 --> 00:18:56,180 So if factors two, x minus seven, x plus seven. 259 00:19:05,840 --> 00:19:07,280 And now what? 260 00:19:09,380 --> 00:19:09,620 Zero. 261 00:19:10,620 --> 00:19:11,200 Zero. 262 00:19:14,780 --> 00:19:18,940 X minus seven equals zero, x plus seven equals zero. 263 00:19:19,780 --> 00:19:21,940 X equals seven, x equals negative seven. 264 00:19:29,120 --> 00:19:30,080 What? 265 00:19:33,060 --> 00:19:33,360 No. 266 00:19:44,960 --> 00:19:45,540 What? 267 00:19:52,280 --> 00:19:53,920 This is not the difference of squares. 268 00:19:54,080 --> 00:19:54,660 What is this? 269 00:19:55,980 --> 00:19:56,460 Sum of squares. 270 00:19:58,740 --> 00:20:00,640 Do sum of squares factor? 271 00:20:01,860 --> 00:20:01,940 No. 272 00:20:05,940 --> 00:20:06,800 Doesn't factor. 273 00:20:06,800 --> 00:20:09,340 That is not a factor. 274 00:20:13,980 --> 00:20:15,300 What does that mean? 275 00:20:18,780 --> 00:20:19,900 There's no solutions. 276 00:20:21,080 --> 00:20:22,060 There is a solution set. 277 00:20:22,240 --> 00:20:22,660 What is it? 278 00:20:23,660 --> 00:20:23,880 Empty set. 279 00:20:24,100 --> 00:20:24,660 The empty set. 280 00:20:25,160 --> 00:20:26,100 Either that or that. 281 00:20:27,300 --> 00:20:27,420 Okay. 282 00:20:28,600 --> 00:20:34,320 Just because a polynomial doesn't factor doesn't mean there's no solutions. 283 00:20:34,680 --> 00:20:35,400 We'll get to that later. 284 00:20:36,400 --> 00:20:38,460 But this one definitely does not factor. 285 00:20:39,640 --> 00:20:40,160 There's no solutions. 286 00:20:43,980 --> 00:20:45,840 You could also think this way. 287 00:20:46,280 --> 00:20:50,220 X squared equals negative twenty-five. 288 00:20:51,120 --> 00:20:55,003 There's no way or something squared which will always be positive to equal a 289 00:20:55,003 --> 00:20:55,820 negative number. 290 00:20:56,280 --> 00:20:57,740 There's definitely no solutions here. 291 00:21:18,100 --> 00:21:19,820 Is this the difference of squares? 292 00:21:22,760 --> 00:21:22,800 Yeah. 293 00:21:23,840 --> 00:21:24,340 It is. 294 00:21:30,400 --> 00:21:34,080 Two x, squared, is four x squared. 295 00:21:34,360 --> 00:21:35,260 What squared is 5? 296 00:21:35,820 --> 00:21:36,340 Square root of 5. 297 00:21:37,600 --> 00:21:40,400 Square root of 5. 298 00:21:51,840 --> 00:21:52,800 And now what? 299 00:21:55,960 --> 00:21:56,480 ZPP. 300 00:21:57,420 --> 00:21:58,980 We have a product that equals zero. 301 00:21:59,200 --> 00:22:01,340 We set each factor equal to zero. 302 00:22:30,860 --> 00:22:35,720 And we got negative the square root of 5 over 2. 303 00:22:36,460 --> 00:22:38,620 And the square root of 5 over 2. 304 00:22:42,280 --> 00:22:46,182 Sometimes when we have the same number but just opposite signs, well it's not the 305 00:22:46,182 --> 00:22:46,760 same number, 306 00:22:46,760 --> 00:22:48,040 but opposite numbers. 307 00:22:49,760 --> 00:22:51,100 We sometimes write it like this. 308 00:22:51,440 --> 00:22:54,380 Plus and minus the square root of 5 over 2. 309 00:22:58,440 --> 00:23:01,340 Like we could have done that for this one. 310 00:23:02,420 --> 00:23:05,980 We could have wrote plus and minus 7 which is two different numbers. 311 00:23:22,940 --> 00:23:24,420 Yeah, I think we got time. 312 00:23:32,340 --> 00:23:34,320 We had this equation before. 313 00:23:34,520 --> 00:23:34,660 Yes. 314 00:23:37,780 --> 00:23:38,280 Okay. 315 00:23:44,440 --> 00:23:48,360 I agree this is the equivalent equation. 316 00:23:48,940 --> 00:23:49,320 Yes. 317 00:23:55,280 --> 00:23:58,720 Can I just take the square root of both sides? 318 00:24:05,280 --> 00:24:05,420 Okay. 319 00:24:07,880 --> 00:24:08,340 Possibly. 320 00:24:10,820 --> 00:24:16,400 But then we only get one solution. 321 00:24:21,520 --> 00:24:24,240 So just taking the square root of both sides, you only get one solution. 322 00:24:27,740 --> 00:24:28,620 Okay. 323 00:24:36,280 --> 00:24:40,420 However, we know this can be factored, right? 324 00:24:41,820 --> 00:24:48,280 This can be factored to X minus 7 times X plus 7. 325 00:24:49,020 --> 00:25:02,900 And then by the ZPP, we get X minus 7 equals 0 and X plus 7 equals 0. 326 00:25:03,040 --> 00:25:06,000 And we got our two solutions, negative 7 and 7. 327 00:25:07,680 --> 00:25:08,480 Okay. 328 00:25:13,560 --> 00:25:19,540 So if X squared equals any number, let's call it A. 329 00:25:20,340 --> 00:25:22,620 A has to be greater than or equal to 0. 330 00:25:27,700 --> 00:25:32,540 X squared minus A equals 0 is equivalent. 331 00:25:41,700 --> 00:25:43,260 Is this the difference of squares? 332 00:25:44,960 --> 00:25:45,260 Yeah. 333 00:25:46,520 --> 00:25:54,431 X squared minus the square root of a squared factors to x minus the square root of 334 00:25:54,431 --> 00:25:58,000 a, times x plus the square root of a. 335 00:26:01,100 --> 00:26:04,200 And by the zero product property. 336 00:26:08,040 --> 00:26:18,440 X minus the square root of a equals zero, so x equals the square root of a. 337 00:26:19,280 --> 00:26:21,360 And X equals negative square root of A. 338 00:26:22,800 --> 00:26:29,000 And we get negative the square root of a. 339 00:26:34,420 --> 00:26:39,820 Which could be written as plus or minus the square root of A. 340 00:26:40,380 --> 00:26:43,580 And that number, number A does not matter. 341 00:26:44,120 --> 00:26:46,480 As long as a positive number. 342 00:26:47,440 --> 00:26:48,680 Why is the number of the negative number? 343 00:26:51,260 --> 00:26:52,060 What'd you say? 344 00:26:55,600 --> 00:26:59,780 This works always, except for when A is a negative number. 345 00:27:00,960 --> 00:27:01,200 It does. 346 00:27:02,680 --> 00:27:06,480 The square root of negative numbers don't exist. 347 00:27:07,560 --> 00:27:08,560 There's no real numbers. 348 00:27:10,020 --> 00:27:10,140 Okay. 349 00:27:10,900 --> 00:27:22,778 So if any time we have X squared equals a positive number, we know the two solutions 350 00:27:22,778 --> 00:27:29,000 are X equals the square root of that number. 351 00:27:30,220 --> 00:27:32,780 And X equals negative the square root of that number. 352 00:27:33,520 --> 00:27:34,980 We just proved it. 353 00:27:35,880 --> 00:27:36,400 Okay. 354 00:27:36,420 --> 00:27:38,320 We got these two equations. 355 00:27:40,720 --> 00:27:40,740 Okay. 356 00:27:41,400 --> 00:27:42,720 And we often write it like this. 357 00:27:42,960 --> 00:27:45,680 X equals plus or minus the square root of that number. 358 00:27:50,720 --> 00:27:51,240 Okay. 359 00:27:51,480 --> 00:27:55,700 So if this is true, then this is true. 360 00:27:55,820 --> 00:27:57,040 Let's write that down. 361 00:27:58,440 --> 00:28:08,832 If X squared equals a number, as long as positive, then X equals plus and minus the 362 00:28:08,832 --> 00:28:10,960 square root of A. 363 00:28:11,420 --> 00:28:12,640 Which is two equations. 364 00:28:12,940 --> 00:28:14,220 X equals the square root of A. 365 00:28:15,640 --> 00:28:17,980 And X equals the negative square root of A. 366 00:28:20,640 --> 00:28:21,280 Okay. 367 00:28:21,660 --> 00:28:24,000 This is called the square root principle. 368 00:28:36,860 --> 00:28:43,420 And it's going to allow us to solve quadratic equations without factoring. 369 00:28:44,000 --> 00:28:45,860 Because we just proved it always factors. 370 00:28:47,180 --> 00:28:52,420 So for example, if we have X squared equals 25. 371 00:28:57,520 --> 00:29:00,380 It looks like you're taking the square root of both sides, but we're not. 372 00:29:01,560 --> 00:29:03,240 It's by the square root principle. 373 00:29:15,960 --> 00:29:19,740 That X will equal plus or minus the square root of 25. 374 00:29:20,940 --> 00:29:21,500 Okay. 375 00:29:21,860 --> 00:29:23,800 We're not taking the square root of both sides. 376 00:29:25,200 --> 00:29:27,200 We're using the square root principle. 377 00:29:27,320 --> 00:29:31,878 Anytime something squared equals a number, whatever is squared equals plus or minus 378 00:29:31,878 --> 00:29:33,580 the square root of that number. 379 00:29:33,800 --> 00:29:35,220 Which is two separate equations. 380 00:29:36,500 --> 00:29:37,020 Okay. 381 00:29:38,520 --> 00:29:39,220 Plus or minus five. 382 00:29:40,140 --> 00:29:41,900 And that gives our two solutions. 383 00:29:46,220 --> 00:29:46,860 Okay. 384 00:29:49,300 --> 00:29:56,020 So, anytime you have X squared, let's just do another one. 385 00:30:01,280 --> 00:30:02,560 Equals 50. 386 00:30:06,660 --> 00:30:09,720 We're not taking the square root of both sides. 387 00:30:09,960 --> 00:30:11,380 We're going to use the what principle? 388 00:30:12,940 --> 00:30:14,160 Square root principle. 389 00:30:14,660 --> 00:30:17,400 Square root principle. 390 00:30:20,660 --> 00:30:25,600 Then X will equal plus or minus the square root of 50. 391 00:30:27,920 --> 00:30:29,680 And what does that simplify to? 392 00:30:31,180 --> 00:30:32,720 Five square root of two. 393 00:30:34,100 --> 00:30:34,860 Five square root of two. 394 00:30:37,000 --> 00:30:38,760 We have our two solutions. 395 00:30:41,380 --> 00:30:41,860 Okay. 396 00:30:42,620 --> 00:30:46,934 All the square root principle is saving time where we're not bringing this to the 397 00:30:46,934 --> 00:30:47,520 other side. 398 00:30:48,540 --> 00:30:49,700 And we're not factoring. 399 00:30:56,440 --> 00:30:58,880 And we use this as ZPP. 400 00:31:12,740 --> 00:31:13,380 Okay. 401 00:31:22,660 --> 00:31:24,520 We got our two solutions. 402 00:31:25,720 --> 00:31:26,400 Our two equations. 403 00:31:27,340 --> 00:31:27,360 Okay. 404 00:31:28,020 --> 00:31:34,086 So, because we know it's going to be the difference of squares, there's no linear 405 00:31:34,086 --> 00:31:34,460 term. 406 00:31:35,800 --> 00:31:36,440 Okay. 407 00:31:36,600 --> 00:31:39,920 It will factor and we'll end up with these two equations. 408 00:31:40,780 --> 00:31:41,180 Okay. 409 00:31:41,300 --> 00:31:42,960 And we write this all in one equation. 410 00:31:44,420 --> 00:31:44,560 Right. 411 00:31:44,700 --> 00:31:44,940 We could. 412 00:31:48,180 --> 00:31:48,580 Okay. 413 00:31:48,940 --> 00:31:50,020 So, that is the square root principle. 414 00:31:50,300 --> 00:31:54,120 We can only do it when we have something squared equals some positive number. 415 00:31:58,100 --> 00:31:58,500 Okay. 416 00:31:58,620 --> 00:31:59,840 And you go right to this equation. 417 00:32:01,360 --> 00:32:02,040 Does this make sense? 418 00:32:03,300 --> 00:32:03,600 Yes. 419 00:32:04,740 --> 00:32:08,100 This is going to be very important for the next thing we learn later. 420 00:32:12,980 --> 00:32:13,460 Okay. 421 00:32:13,780 --> 00:32:15,600 I think we could stop here. 422 00:32:16,560 --> 00:32:19,600 Most of the problems, we're not going to be able to use the square root principle. 423 00:32:20,180 --> 00:32:24,180 We just have to factor and use what property? 424 00:32:26,720 --> 00:32:27,880 Zero product property. 425 00:32:28,600 --> 00:32:30,520 And you need to write in there where you use it. 426 00:32:31,060 --> 00:32:33,360 You write two separate equations and then solve.