1 00:00:05,000 --> 00:00:09,480 Right this now we're solving a quadratic 2 00:00:11,920 --> 00:00:14,260 Equation by factoring 3 00:00:17,300 --> 00:00:21,580 How do we so what's the first step we always do 4 00:00:24,840 --> 00:00:25,380 No 5 00:00:29,300 --> 00:00:32,360 Well before we do any of that, okay 6 00:00:32,940 --> 00:00:33,520 Oh 7 00:00:34,740 --> 00:00:49,980 We have to have a zero on one side, so what do I do subtract the 25? Yes 8 00:00:49,980 --> 00:00:51,900 Now what 9 00:00:55,840 --> 00:00:56,280 Oh 10 00:01:00,400 --> 00:01:03,060 Well in order to factor this what do we do first 11 00:01:08,200 --> 00:01:10,140 Multiply this out 12 00:01:10,140 --> 00:01:11,520 Oh 13 00:01:19,820 --> 00:01:21,860 And we got to distribute some stuff 14 00:01:40,160 --> 00:01:40,680 I 15 00:01:42,320 --> 00:01:43,660 Was that my 16 16 00:01:47,040 --> 00:01:47,560 Okay 17 00:01:48,420 --> 00:01:54,032 Yeah, we got to get zero to one on one of the two sides and then we got to put it in 18 00:01:54,032 --> 00:01:54,900 standard form 19 00:01:55,720 --> 00:01:57,020 standard form is 20 00:01:57,680 --> 00:01:59,500 some number of times x squared 21 00:02:00,260 --> 00:02:02,820 Plus the linear term plus the constant term 22 00:02:04,640 --> 00:02:06,840 We'll talk about that 23 00:02:07,640 --> 00:02:11,380 Now what? Now we have to factor. 24 00:02:13,340 --> 00:02:15,560 Got it, right 25 00:02:18,460 --> 00:02:23,040 What is it x plus 2 26 00:02:28,740 --> 00:02:31,000 And then what do we use 27 00:02:33,620 --> 00:02:35,880 Zero product property 28 00:02:36,800 --> 00:02:39,240 We have a product that equals zero 29 00:02:39,800 --> 00:02:42,520 Either x minus eight equals zero, or 30 00:02:43,300 --> 00:02:48,120 x plus 2 equals zero and now we no longer have a 31 00:02:48,940 --> 00:02:52,640 Quadratic equation we have two linear equations 32 00:02:53,640 --> 00:02:56,220 Thanks to the zero product property 33 00:02:59,300 --> 00:03:03,980 We got our two solutions 34 00:03:09,080 --> 00:03:14,865 This makes sense we did this last week the only thing we probably didn't do yeah was 35 00:03:14,865 --> 00:03:16,380 multiply something out 36 00:03:22,160 --> 00:03:25,480 That's solving by factoring: get everything to one side. 37 00:03:27,420 --> 00:03:29,840 Make sure zero is on the other side really 38 00:03:30,920 --> 00:03:34,840 Then we put in a standard form we factor it use zero product property and so on 39 00:03:35,740 --> 00:03:37,140 All right. Next question 40 00:03:56,880 --> 00:03:57,480 So 41 00:04:00,080 --> 00:04:04,000 By factoring we would get this to this side, right? 42 00:04:05,900 --> 00:04:07,920 And then how do we factor that 43 00:04:08,900 --> 00:04:09,720 Oh 44 00:04:09,720 --> 00:04:16,256 Difference of squares, okay. However, this is missing the linear term, right? So 45 00:04:16,256 --> 00:04:18,380 which property can we use? 46 00:04:20,640 --> 00:04:22,900 The SRP which stands for 47 00:04:23,900 --> 00:04:25,900 Square root principle. 48 00:04:27,860 --> 00:04:28,560 Square root. 49 00:04:30,440 --> 00:04:31,500 Principle. 50 00:04:36,100 --> 00:04:40,800 You equals plus or minus the square root of 25 51 00:04:44,980 --> 00:04:49,640 You equals plus or minus 5 52 00:04:57,220 --> 00:05:03,780 Which is two equations and we got our two solutions, right? 53 00:05:08,640 --> 00:05:11,300 Five and negative five, but I'm gonna go back for a sec 54 00:05:17,860 --> 00:05:19,580 When can we use the square root principle? 55 00:05:24,120 --> 00:05:25,400 When there's no what 56 00:05:26,260 --> 00:05:27,600 Linear term 57 00:05:28,300 --> 00:05:30,360 sometimes we call the coefficient of the 58 00:05:30,360 --> 00:05:36,040 Quadratic term we call it a the coefficient of the linear term 59 00:05:36,040 --> 00:05:41,734 We call it b, and the constant term we call c. Whenever b is zero, we can use the 60 00:05:41,734 --> 00:05:43,280 square root principle. 61 00:05:43,280 --> 00:05:46,320 What I failed to tell you was 62 00:05:47,960 --> 00:05:52,400 You actually equals X minus 3 63 00:05:56,500 --> 00:05:57,340 Okay 64 00:05:59,620 --> 00:06:00,460 Okay 65 00:06:01,020 --> 00:06:08,060 Okay, so we got X minus 3 squared equals 25 66 00:06:13,320 --> 00:06:15,980 Can we still use the square root principle 67 00:06:19,280 --> 00:06:20,960 We use it right here 68 00:06:22,680 --> 00:06:31,580 Here, x minus three equals plus or minus the square root of twenty-five. 69 00:06:36,220 --> 00:06:40,080 X minus 3 is plus or minus 5 70 00:06:42,440 --> 00:06:46,540 X is 3 plus or minus 5 71 00:06:46,540 --> 00:06:53,380 X equals 3 plus 5 or X equals 3 minus 5 72 00:06:53,380 --> 00:06:58,872 Okay, that's a great question why did we not do that right here one is got told you 73 00:06:58,872 --> 00:07:00,460 just solved by factoring 74 00:07:02,160 --> 00:07:04,540 Two is the whole point is 75 00:07:05,840 --> 00:07:09,900 To show us even though this is not X squared equals the number 76 00:07:10,920 --> 00:07:12,280 This is something 77 00:07:12,980 --> 00:07:14,160 squared equals a number 78 00:07:16,240 --> 00:07:18,160 Okay, let's call this 79 00:07:19,260 --> 00:07:25,446 You for a second we got u squared equals 25 we could still use the square root 80 00:07:25,446 --> 00:07:26,160 principle 81 00:07:28,220 --> 00:07:30,460 Okay, whatever is squared 82 00:07:35,200 --> 00:07:42,300 Whatever is squared this case is not just X it's X minus 3 we simply go X minus 3 83 00:07:42,940 --> 00:07:48,057 Equals plus or minus the square root of the other side and that is using the square 84 00:07:48,057 --> 00:07:48,920 root principle 85 00:07:51,280 --> 00:07:51,840 Okay 86 00:07:54,880 --> 00:07:56,320 Go with me for a second 87 00:07:57,180 --> 00:07:57,740 Okay 88 00:07:58,440 --> 00:08:02,300 It saves us a lot of time. We don't have to multiply out 89 00:08:02,980 --> 00:08:05,740 and bring everything over and then 90 00:08:05,740 --> 00:08:07,300 factor again 91 00:08:08,700 --> 00:08:11,220 Okay, so let me do another example 92 00:08:11,220 --> 00:08:11,740 We 93 00:08:11,740 --> 00:08:16,760 Have something squared equals a number 94 00:08:18,580 --> 00:08:20,520 Can we use the square root principle 95 00:08:22,620 --> 00:08:25,440 Yeah, whatever is squared let me call it you for a second 96 00:08:27,000 --> 00:08:31,720 X plus five equals plus or minus the square root of nine. 97 00:08:35,220 --> 00:08:39,260 X plus five equals plus or minus three. 98 00:08:41,800 --> 00:08:44,920 X equals five plus or minus three. 99 00:08:46,680 --> 00:08:51,060 Which is X two different equations 100 00:08:56,920 --> 00:08:58,480 We get 8 101 00:09:00,440 --> 00:09:01,040 2 102 00:09:04,040 --> 00:09:08,680 Negative five! Yes, thank you. 103 00:09:13,240 --> 00:09:19,460 Negative 2 and X equals negative 8 we got our two solutions 104 00:09:20,880 --> 00:09:21,840 Oops 105 00:09:31,720 --> 00:09:35,620 Okay, so we could use the square root principle 106 00:09:37,160 --> 00:09:40,720 When anything squared equals a number 107 00:09:41,820 --> 00:09:42,920 whatever squared 108 00:09:44,120 --> 00:09:49,320 Equals plus and minus the square root of the other side, which is two separate 109 00:09:49,320 --> 00:09:51,920 equations, and then we we go from there 110 00:09:51,920 --> 00:09:56,776 We're gonna solve this by well, we're sorry this can easily be solved by factor, 111 00:09:56,776 --> 00:09:57,140 right? 112 00:10:00,340 --> 00:10:02,140 Okay, what's this factor to 113 00:10:08,180 --> 00:10:10,320 Actually, let me do a different example 114 00:10:16,720 --> 00:10:24,280 This equals x plus five times x plus one. Then what do we do? 115 00:10:27,820 --> 00:10:31,020 Zero product property X plus 5 116 00:10:32,320 --> 00:10:34,400 X plus 1 117 00:10:43,500 --> 00:10:51,420 Actually, let me do a different problem 118 00:10:55,320 --> 00:11:01,320 Now let's do with this sorry, okay, let's take the same equation 119 00:11:10,140 --> 00:11:10,700 I 120 00:11:13,340 --> 00:11:15,400 Put this on this side 121 00:11:27,340 --> 00:11:29,880 I'm going to add something to both sides 122 00:11:33,780 --> 00:11:40,120 Okay, so when this factors it's a perfect square 123 00:11:51,020 --> 00:11:54,020 So I'm gonna add some number 124 00:11:56,200 --> 00:12:01,120 So it will factor to the same exact two binomials 125 00:12:03,200 --> 00:12:08,400 This gives a perfect square. What do you think we should add? 126 00:12:11,460 --> 00:12:13,480 Okay, she's saying add 9 127 00:12:22,020 --> 00:12:28,120 If you add 9 we'll talk about a second how we do that what is this factor to 128 00:12:29,100 --> 00:12:30,900 X plus 3 X plus 3 129 00:12:32,260 --> 00:12:33,700 X plus 3 130 00:12:34,420 --> 00:12:37,440 Okay, we'll come back this minute. Let's just finish this 131 00:12:47,080 --> 00:12:49,180 And once we're here 132 00:12:50,060 --> 00:12:55,440 What can we use now? What's something squared equals number square root principle? 133 00:12:55,720 --> 00:12:59,980 We get X plus 3 is plus or minus the square root of 4 134 00:13:07,420 --> 00:13:09,760 Which is two equations 135 00:13:21,260 --> 00:13:25,080 There are two same two solutions we solved by factoring earlier ago 136 00:13:29,480 --> 00:13:32,200 Once we write our two binomials 137 00:13:34,260 --> 00:13:35,840 These two 138 00:13:37,240 --> 00:13:39,300 Have to multiply to be this 139 00:13:42,020 --> 00:13:43,140 And then 140 00:13:44,820 --> 00:13:50,800 These two 3x and 3x have to add to be 6x 141 00:13:52,920 --> 00:13:53,600 Okay 142 00:13:55,680 --> 00:14:02,470 So we literally just take this B value divide by 2 and that's going to be the number 143 00:14:02,470 --> 00:14:03,440 here in here 144 00:14:06,440 --> 00:14:09,500 Okay, when we factor it which is B over 2 145 00:14:12,760 --> 00:14:13,800 And then 146 00:14:17,760 --> 00:14:21,020 To figure out what number we add we just take B over 147 00:14:22,140 --> 00:14:27,720 2 and we square it and that's the number we add so 6 over 2 148 00:14:28,820 --> 00:14:34,420 Squared this 3 squared is 9 and that's what we add to both sides of the equation 149 00:14:36,980 --> 00:14:40,400 Okay, this is called solving 150 00:14:44,080 --> 00:14:45,600 By completing 151 00:14:49,220 --> 00:14:50,540 The square 152 00:14:53,200 --> 00:15:00,560 Okay, why is it called solving by completing this square because once we're here 153 00:15:02,980 --> 00:15:09,524 We have to make we have to add something to make it a perfect square so we need to 154 00:15:09,524 --> 00:15:11,360 complete the square and 155 00:15:11,360 --> 00:15:14,300 That's we always add B over 2 squared 156 00:15:15,080 --> 00:15:19,760 And we got this we now completed this square. It's a perfect square when it factors 157 00:15:19,760 --> 00:15:22,520 It's gonna be something square a binomial squared 158 00:15:25,420 --> 00:15:28,600 Okay, let's try another one 159 00:15:28,600 --> 00:15:30,600 Here's a quadratic 160 00:15:34,280 --> 00:15:37,220 Polynomial that does not factor 161 00:15:39,420 --> 00:15:40,140 Okay 162 00:15:42,040 --> 00:15:44,980 Well, it actually does we'll talk about that in a second 163 00:15:46,700 --> 00:15:49,320 But the only options we know of our what? 164 00:15:50,960 --> 00:15:53,120 Negative 5 and 1 or 165 00:15:53,120 --> 00:15:57,220 five and negative one, and neither pair adds to eight. 166 00:15:58,020 --> 00:16:00,160 We agree. Yes. Okay 167 00:16:01,400 --> 00:16:04,420 So this cannot be solved by factoring 168 00:16:06,280 --> 00:16:12,080 The only other way we know of is by completing the square so the first step 169 00:16:16,880 --> 00:16:23,160 We get the constant term on the right the quadratic and the linear on the left 170 00:16:26,500 --> 00:16:28,380 And now we got to add something 171 00:16:32,840 --> 00:16:34,680 So becomes a perfect square 172 00:16:37,280 --> 00:16:40,700 You got to complete the square, so what am I gonna add? 173 00:16:42,350 --> 00:16:44,890 These two numbers have to add to eight. 174 00:16:46,690 --> 00:16:47,830 So which is 175 00:16:49,370 --> 00:16:51,190 8 over 2 is 176 00:16:52,270 --> 00:16:55,210 4 so it's plus 4 plus 4 177 00:16:56,110 --> 00:17:00,790 And then to figure out they have to multiply to be this number so we square 178 00:17:02,530 --> 00:17:03,130 16 179 00:17:04,770 --> 00:17:08,390 We add sixteen to both sides. 180 00:17:21,070 --> 00:17:25,470 And now we have something squared equals a number. What can we use? 181 00:17:26,710 --> 00:17:28,430 Square root principle 182 00:17:31,310 --> 00:17:35,990 X plus four equals plus or minus the square root of twenty-one. 183 00:17:36,990 --> 00:17:40,750 Does the square root of twenty-one simplify? No. 184 00:17:42,730 --> 00:17:43,310 Okay 185 00:17:50,750 --> 00:17:54,350 Okay, and those are the two solutions 186 00:17:55,310 --> 00:17:56,990 It's negative 4 187 00:17:56,990 --> 00:17:58,890 plus the square root of twenty-one, 188 00:17:59,790 --> 00:18:02,690 and negative four minus the square root of twenty-one. 189 00:18:03,330 --> 00:18:06,210 Both solutions are irrational numbers 190 00:18:10,170 --> 00:18:11,770 But that's it 191 00:18:12,850 --> 00:18:15,450 So we were able to solve this 192 00:18:16,710 --> 00:18:19,250 Even though it could not be solved by factoring 193 00:18:22,430 --> 00:18:24,730 Okay, and let me show you something 194 00:18:30,690 --> 00:18:31,870 This 195 00:18:31,870 --> 00:18:34,170 Actually does factor 196 00:18:36,810 --> 00:18:40,770 This would be x and x, right? It's simply 197 00:18:45,650 --> 00:18:46,330 For 198 00:18:48,050 --> 00:18:53,504 four plus the square root of twenty-one, and four minus the square root of twenty- 199 00:18:53,504 --> 00:18:53,770 one. 200 00:18:54,730 --> 00:18:55,790 So it's fine 201 00:18:59,610 --> 00:19:03,050 Okay, and there's no way we were gonna guess that right 202 00:19:03,650 --> 00:19:06,270 The only we guess is integer values 203 00:19:07,250 --> 00:19:11,350 Okay, and these these two numbers have the same property 204 00:19:12,090 --> 00:19:15,730 they multiply to be negative 5 and 205 00:19:17,590 --> 00:19:18,950 They add to be 8 206 00:19:20,470 --> 00:19:25,090 Okay, they're just not nice whole numbers or integers I should say 207 00:19:27,250 --> 00:19:32,968 These add to eight: four plus the square root of twenty-one plus four minus the 208 00:19:32,968 --> 00:19:34,850 square root of twenty-one. 209 00:19:34,850 --> 00:19:36,730 That's 0 we get 8 210 00:19:37,590 --> 00:19:39,810 And do they multiply to be negative 5? 211 00:19:41,730 --> 00:19:44,070 They do let's try it 212 00:19:53,750 --> 00:19:54,630 16 213 00:19:56,490 --> 00:20:00,740 Minus four times the square root of twenty-one, plus four times the square root of 214 00:20:00,740 --> 00:20:01,310 twenty-one, 215 00:20:04,310 --> 00:20:05,190 Minus 216 00:20:05,830 --> 00:20:06,330 21 217 00:20:07,430 --> 00:20:11,010 These terms cancel, and sixteen minus twenty-one gives negative five. 218 00:20:11,790 --> 00:20:14,970 It's kind of cool. That's pretty cool. It is pretty cool 219 00:20:20,750 --> 00:20:24,410 Okay, let's try that one and then we'll get to the more difficult 220 00:20:24,670 --> 00:20:27,110 Again there's no way that's 221 00:20:27,710 --> 00:20:32,070 Solve this by factoring. So we're gonna complete the square first step 222 00:20:36,010 --> 00:20:45,450 We gotta get the constant over here and then we complete the square 223 00:20:52,790 --> 00:20:57,170 We take this number divide by 2 and square 224 00:21:07,850 --> 00:21:09,910 We add twenty-five to both sides. 225 00:21:19,590 --> 00:21:21,450 And this definitely factors to 226 00:21:23,430 --> 00:21:26,890 Half of that number would be over 2 so that's plus 5 plus 5 227 00:21:37,210 --> 00:21:41,390 And now what square root principle 228 00:21:45,890 --> 00:21:50,990 X plus five equals plus or minus the square root of twenty-eight. 229 00:22:11,270 --> 00:22:14,470 Does the square root of twenty-eight simplify? I should have done that first. 230 00:22:21,890 --> 00:22:22,490 Oh 231 00:22:22,490 --> 00:22:28,490 There we go, we got our two solutions 232 00:22:33,870 --> 00:22:38,670 We can now solve quadratic equations when they cannot be factored over the integers, 233 00:22:38,670 --> 00:22:40,230 or at least are not obviously factorable. 234 00:22:44,830 --> 00:22:46,450 Let's try that one 235 00:22:46,450 --> 00:22:49,930 Okay, what's the first thing we do? 236 00:22:54,070 --> 00:22:55,810 We got to get the constant on the right 237 00:23:04,150 --> 00:23:06,130 Okay, and we have a little problem on this one 238 00:23:10,050 --> 00:23:12,710 We want to get x plus or minus some number 239 00:23:18,750 --> 00:23:20,650 We want just an x here 240 00:23:22,670 --> 00:23:26,070 We could use square root of two times square root of two, 241 00:23:27,870 --> 00:23:30,890 Because then that does multiply to be 2x squared 242 00:23:31,810 --> 00:23:33,430 But it's way too complicated 243 00:23:35,070 --> 00:23:36,430 Okay, so 244 00:23:37,150 --> 00:23:39,850 Whenever the leading coefficients not one 245 00:23:39,850 --> 00:23:44,785 We have to divide both sides by the leading coefficient. We need to change the 246 00:23:44,785 --> 00:23:46,430 leading coefficient to one 247 00:23:47,310 --> 00:23:49,890 So we divide by 2 divided by 2 248 00:23:50,930 --> 00:23:54,770 We get x squared minus 3 over 2x 249 00:23:55,510 --> 00:23:57,070 5 over 2 250 00:24:10,750 --> 00:24:13,870 And now we complete the square 251 00:24:17,510 --> 00:24:23,290 And now we're gonna deal with some fractions and that's okay, this is my B value 252 00:24:24,670 --> 00:24:28,530 Negative three over two. Take b over two, squared. 253 00:24:32,870 --> 00:24:37,970 Which is negative 3 over 4 squared which is 254 00:24:44,650 --> 00:24:46,170 9 16th 255 00:24:46,170 --> 00:24:47,830 So we add 9 16th 256 00:25:01,730 --> 00:25:05,090 It's that factors to x plus or minus 257 00:25:06,030 --> 00:25:08,130 Minus minus what? 258 00:25:09,650 --> 00:25:16,111 3 over 4 it's the number of before we square it or it's also the square of that but 259 00:25:16,111 --> 00:25:18,290 then you got to take care of 260 00:25:18,290 --> 00:25:18,850 the sign 261 00:25:25,910 --> 00:25:29,370 Okay, and then we got to get a common denominator over here 262 00:25:35,190 --> 00:25:39,890 That's 40 plus 9 49 over 16 263 00:25:41,670 --> 00:25:45,570 That right, yes 264 00:25:52,570 --> 00:25:53,690 Now what 265 00:25:53,690 --> 00:26:01,553 Something squared equals a number. Use the square root principle: x minus three 266 00:26:01,553 --> 00:26:02,250 fourths 267 00:26:03,520 --> 00:26:08,700 equals plus or minus the square root of 49 16th 268 00:26:16,680 --> 00:26:20,260 Was that 7 over 4? 269 00:26:50,340 --> 00:26:58,400 Which is 10 fourths or negative 4 fourths? 270 00:27:04,300 --> 00:27:06,500 which is five halves. Yes. 271 00:27:20,000 --> 00:27:24,543 This happened to be a perfect square, so we ended up with no square roots. When that 272 00:27:24,543 --> 00:27:26,220 number is not a perfect square, 273 00:27:27,340 --> 00:27:30,180 You're gonna end up with square roots in your solution 274 00:27:37,580 --> 00:27:42,320 We're gonna do it 275 00:27:42,320 --> 00:27:43,460 You try one more 276 00:27:43,460 --> 00:27:53,820 Actually, what's the first step? Okay, that's good 277 00:27:57,940 --> 00:28:03,580 Yeah, man, what's the next step? Okay, you guys are paying attention I like this 278 00:28:13,740 --> 00:28:15,880 Divide what by two 279 00:28:17,760 --> 00:28:20,060 Yeah, we sometimes call that the B value 280 00:28:39,500 --> 00:28:44,720 right, we got to add it to complete the square 281 00:28:57,680 --> 00:28:59,080 Oh 282 00:28:59,080 --> 00:29:13,460 Six goes into 144 24. Yes 283 00:29:13,460 --> 00:29:20,780 What's five times 24 120 plus 49 is 284 00:29:27,460 --> 00:29:27,820 One hundred sixty-nine. 285 00:29:38,020 --> 00:29:42,780 Common say we can't do that because uh, this is 13 286 00:29:43,620 --> 00:29:48,340 Times 13 and this is 12 times 12, but those don't simplify 287 00:29:50,440 --> 00:29:53,840 But you might be saying something different 288 00:29:55,140 --> 00:29:58,860 We use the square root principle. 289 00:29:58,860 --> 00:30:03,083 X minus seven twelfths equals plus or minus the square root of one hundred sixty- 290 00:30:03,083 --> 00:30:06,420 nine over one hundred forty-four. You were ahead of me, I think. 291 00:30:10,060 --> 00:30:16,620 Okay, clear it now what 292 00:30:32,200 --> 00:30:33,600 I 293 00:30:33,600 --> 00:30:35,300 Like that 294 00:30:39,800 --> 00:30:43,660 Negative six twelves 295 00:30:49,500 --> 00:30:52,700 Twenty twelfths which is 296 00:30:56,100 --> 00:30:56,960 Five thirds 297 00:31:17,340 --> 00:31:18,180 Oops 298 00:31:18,180 --> 00:31:21,460 Not an ordered pair: the solution set. 299 00:31:29,520 --> 00:31:35,980 That was luck it does tell us whenever you get fractions here well 300 00:31:40,500 --> 00:31:42,520 What was the original equation 301 00:31:46,640 --> 00:31:47,380 Okay 302 00:32:05,840 --> 00:32:06,720 This 303 00:32:06,720 --> 00:32:09,180 This does not factor to this why not 304 00:32:10,060 --> 00:32:10,660 Because 305 00:32:12,740 --> 00:32:13,340 Because 306 00:32:15,100 --> 00:32:19,380 The leading term would not be six x squared, correct? 307 00:32:20,540 --> 00:32:23,900 However, it's right to X times X is not six X square 308 00:32:24,540 --> 00:32:30,080 However, all you got to do is stick a six out here now. It does factor into that 309 00:32:30,080 --> 00:32:36,240 And if you change this to two times three 310 00:32:39,760 --> 00:32:40,600 Like 311 00:32:46,240 --> 00:32:48,340 You distributed the 312 00:32:49,580 --> 00:32:55,880 Two to this guy you get two X plus one and 313 00:32:57,320 --> 00:32:59,080 Three into this factor gives three 314 00:33:00,800 --> 00:33:01,640 X 315 00:33:01,640 --> 00:33:07,700 So we actually could solve this by factoring Wow 316 00:33:09,440 --> 00:33:09,940 But 317 00:33:11,560 --> 00:33:12,800 That's kind of crazy 318 00:33:13,800 --> 00:33:19,572 Okay, all right two numbers that multiply to be negative 30 add to be negative 7 or 319 00:33:19,572 --> 00:33:19,920 what? 320 00:33:32,160 --> 00:33:35,260 So we could have solved this by factoring 321 00:33:35,260 --> 00:33:37,300 changes to 6x squared 322 00:33:37,900 --> 00:33:44,280 Minus 10x plus 3x minus 5 and a factor by grouping you would have ended up with this 323 00:33:44,800 --> 00:33:46,660 Wow, okay 324 00:33:47,940 --> 00:33:52,560 But we didn't need to solve by completing square any 325 00:33:54,540 --> 00:33:57,920 Any quadratic equation can be solved by completing the square. 326 00:34:00,400 --> 00:34:04,460 Once in a while and give you an example 327 00:34:15,100 --> 00:34:15,460 Let's 328 00:34:15,460 --> 00:34:16,680 Try this one real quick 329 00:34:19,020 --> 00:34:21,100 We'll be the first step 330 00:34:23,800 --> 00:34:28,580 Subtract five on both sides. That's right. 331 00:34:29,410 --> 00:34:34,650 That's right: b over two, squared. Two over two, 332 00:34:34,650 --> 00:34:34,690 squared 333 00:34:35,810 --> 00:34:38,490 1 squared 1 so we add 1 to both sides 334 00:34:51,570 --> 00:34:52,990 Then what do we use 335 00:34:56,710 --> 00:35:03,930 Square root principle: x plus one equals plus or minus the square root of negative 336 00:35:03,930 --> 00:35:04,370 four. 337 00:35:07,170 --> 00:35:10,290 And what is the square root of negative 4 338 00:35:13,470 --> 00:35:16,870 It's not a real number, okay 339 00:35:19,290 --> 00:35:20,370 Not real 340 00:35:21,450 --> 00:35:28,490 There are no real numbers whose square is negative four. 341 00:35:29,610 --> 00:35:31,770 So what does that tell us about the original? 342 00:35:32,730 --> 00:35:33,070 equation 343 00:35:34,510 --> 00:35:35,690 There are no real solutions. 344 00:35:36,490 --> 00:35:39,490 We often say real because we're gonna learn something else later 345 00:35:42,350 --> 00:35:45,730 Does it have a solution set yes, yes it is 346 00:35:47,350 --> 00:35:48,450 The empty set 347 00:35:50,370 --> 00:35:50,650 Okay 348 00:35:51,770 --> 00:35:57,748 So if once you use the square root principle and the right hand side is a negative 349 00:35:57,748 --> 00:35:58,550 you're done 350 00:35:59,390 --> 00:36:00,650 No real solutions 351 00:36:01,410 --> 00:36:04,630 Right and to think about it. There's nothing that you could square 352 00:36:05,250 --> 00:36:08,330 A real square is nonnegative, so it cannot equal this negative number.