1 00:00:00,000 --> 00:00:00,490 2 00:00:00,490 --> 00:00:03,710 In this video, we're going to start having some experience 3 00:00:03,710 --> 00:00:08,400 solving radical equations or equations that involve square 4 00:00:08,400 --> 00:00:11,609 roots or maybe even higher-power roots, but we're 5 00:00:11,609 --> 00:00:14,679 going to also try to understand an interesting 6 00:00:14,679 --> 00:00:17,170 phenomena that occurs when we do these equations. 7 00:00:17,170 --> 00:00:18,840 Let me show you what I'm talking about. 8 00:00:18,839 --> 00:00:23,789 Let's say I have the equation the square root of x is equal 9 00:00:23,789 --> 00:00:27,899 to 2 times x minus 6. 10 00:00:27,899 --> 00:00:29,709 Now, one of the things you're going to see whenever we do 11 00:00:29,710 --> 00:00:32,890 these radical equations is we want to isolate at least one 12 00:00:32,890 --> 00:00:33,570 of the radicals. 13 00:00:33,570 --> 00:00:35,609 There's only one of them in this equation. 14 00:00:35,609 --> 00:00:37,810 And when you isolate one of the radicals on one side of 15 00:00:37,810 --> 00:00:39,700 the equation, this one starts off like that, I have the 16 00:00:39,700 --> 00:00:42,520 square root of x isolated on the left-hand side, then we 17 00:00:42,520 --> 00:00:45,480 square both sides of the equation. 18 00:00:45,479 --> 00:00:48,750 So let us square both sides of the equation. 19 00:00:48,750 --> 00:00:49,929 So I'll just rewrite it again. 20 00:00:49,929 --> 00:00:51,270 We'll do this one slowly. 21 00:00:51,270 --> 00:00:55,480 I'm going to square that and that's going to be equal to 2x 22 00:00:55,479 --> 00:00:57,839 minus 6 squared. 23 00:00:57,840 --> 00:00:59,930 And squaring seems like a valid operation. 24 00:00:59,929 --> 00:01:04,239 If that is equal to that, then that squared should also be 25 00:01:04,239 --> 00:01:06,400 equal to that squared. 26 00:01:06,400 --> 00:01:08,560 So we just keep on going. 27 00:01:08,560 --> 00:01:11,549 So when you take the square root of x and you square it, 28 00:01:11,549 --> 00:01:13,209 that'll just be x. 29 00:01:13,209 --> 00:01:19,280 And we get x is equal to-- this squared is going to be 2x 30 00:01:19,280 --> 00:01:22,340 squared, which is 4x squared. 31 00:01:22,340 --> 00:01:24,560 It's 2x squared, the whole thing. 32 00:01:24,560 --> 00:01:28,030 4x squared, and then you multiply these two, which is 33 00:01:28,030 --> 00:01:29,890 negative 12x. 34 00:01:29,890 --> 00:01:33,920 And then it's going to be twice that, so minus 24x. 35 00:01:33,920 --> 00:01:37,150 And then negative 6 squared is plus 36. 36 00:01:37,150 --> 00:01:41,240 If you found going from this to this difficult, you might 37 00:01:41,239 --> 00:01:44,750 want to review multiplying polynomial expressions or 38 00:01:44,750 --> 00:01:47,019 multiplying binomials, or actually, the special case 39 00:01:47,019 --> 00:01:48,069 where we square binomials. 40 00:01:48,069 --> 00:01:51,579 But the general view, it's this squared, which is that. 41 00:01:51,579 --> 00:01:54,739 And then you have minus 2 times the product of these 2. 42 00:01:54,739 --> 00:01:57,379 The product of those two is minus 12x or negative 12x. 43 00:01:57,379 --> 00:02:00,799 2 times that is negative 24x, and then that squared. 44 00:02:00,799 --> 00:02:03,340 So this is what our equation has I guess we could say 45 00:02:03,340 --> 00:02:06,930 simplified to, and let's see what happens if we subtract x 46 00:02:06,930 --> 00:02:08,710 from both sides of this equation. 47 00:02:08,710 --> 00:02:11,849 So if you subtract x from both sides of this equation, the 48 00:02:11,849 --> 00:02:14,710 left-hand side becomes zero and the right-hand side 49 00:02:14,710 --> 00:02:21,990 becomes 4x squared minus 25x plus 36. 50 00:02:21,990 --> 00:02:25,800 So this radical equation his simplified to just a standard 51 00:02:25,800 --> 00:02:27,670 quadratic equation. 52 00:02:27,669 --> 00:02:30,079 And just for simplicity, not having to worry how to factor 53 00:02:30,080 --> 00:02:31,620 it and grouping and all of that, let's just use the 54 00:02:31,620 --> 00:02:33,230 quadratic formula. 55 00:02:33,229 --> 00:02:37,474 So the quadratic formula tells us that our solutions to this, 56 00:02:37,474 --> 00:02:39,900 x can be negative b. 57 00:02:39,900 --> 00:02:40,990 Negative 25. 58 00:02:40,990 --> 00:02:45,540 The negative of negative 25 is positive 25 plus or minus the 59 00:02:45,539 --> 00:02:49,959 square root of 25 squared. 60 00:02:49,960 --> 00:02:59,200 25 squared is 625, minus 4 times a, which is 4, times c, 61 00:02:59,199 --> 00:03:07,149 which is 36, all of that over 2 times 4, all of that over 8. 62 00:03:07,150 --> 00:03:08,870 So let's get our calculator out to figure out what 63 00:03:08,870 --> 00:03:10,435 this is over here. 64 00:03:10,435 --> 00:03:14,939 65 00:03:14,939 --> 00:03:20,490 So let's say so we have 625 minus-- let's see., this is 66 00:03:20,490 --> 00:03:23,469 going to be 16 times 36. 67 00:03:23,469 --> 00:03:28,280 16 times 36 is equal to 49. 68 00:03:28,280 --> 00:03:28,879 That's nice. 69 00:03:28,879 --> 00:03:29,919 It's a nice perfect square. 70 00:03:29,919 --> 00:03:31,829 We know what the square root of 49 is. 71 00:03:31,830 --> 00:03:32,826 It's 7. 72 00:03:32,825 --> 00:03:34,780 So let me go back to the problem. 73 00:03:34,780 --> 00:03:38,060 So this in here simplified to 49. 74 00:03:38,060 --> 00:03:42,909 So x is equal to 25 plus or minus the square root of 49, 75 00:03:42,909 --> 00:03:46,000 which is 7, all of that over 8. 76 00:03:46,000 --> 00:03:50,960 So our two solutions here, if we add 7, we get x is equal to 77 00:03:50,960 --> 00:03:57,870 25 plus 7 is 32, 32/8, which is equal to 4. 78 00:03:57,870 --> 00:03:59,939 And then our other solution, let me do that 79 00:03:59,939 --> 00:04:01,590 in a different color. 80 00:04:01,590 --> 00:04:07,340 x is equal to 25 minus 7, which is 18/8. 81 00:04:07,340 --> 00:04:12,680 8 goes into 18 two times, remainder 2, so this is equal 82 00:04:12,680 --> 00:04:20,540 to 2 and 2/8 or 2 and 1/4 or 2.25, just like that. 83 00:04:20,540 --> 00:04:22,850 Now, I'm going to show you an interesting 84 00:04:22,850 --> 00:04:24,220 phenomena that occurs. 85 00:04:24,220 --> 00:04:26,480 And maybe you might want to pause it after I show you this 86 00:04:26,480 --> 00:04:28,080 conundrum, although I'm going to tell you why this 87 00:04:28,079 --> 00:04:30,089 conundrum pops up. 88 00:04:30,089 --> 00:04:33,029 Let's try out to see if our solutions actually work. 89 00:04:33,029 --> 00:04:35,559 So let's try x is equal to 4. 90 00:04:35,560 --> 00:04:38,680 If x is equal to 4 works, we get the principal root of 4 91 00:04:38,680 --> 00:04:42,560 should be equal to 2 times 4 minus 6. 92 00:04:42,560 --> 00:04:45,790 The principal root of 4 is positive 2. 93 00:04:45,790 --> 00:04:49,700 Positive 2 should be equal to 2 times 4, which is 8 minus 6, 94 00:04:49,699 --> 00:04:50,420 which it is. 95 00:04:50,420 --> 00:04:51,670 This is true. 96 00:04:51,670 --> 00:04:54,290 So 4 works. 97 00:04:54,290 --> 00:04:57,420 Now, let's try to do the same with 2.25. 98 00:04:57,420 --> 00:05:00,379 According to this, we should be able to take the square 99 00:05:00,379 --> 00:05:05,593 root, the principal root of 2.2-- let me make my radical a 100 00:05:05,593 --> 00:05:06,470 little bit bigger. 101 00:05:06,470 --> 00:05:10,840 The principal root of 2.25 should be equal to 2 times 102 00:05:10,839 --> 00:05:15,619 2.25 minus 6. 103 00:05:15,620 --> 00:05:18,420 Now, you may or may not be able to do this in your head. 104 00:05:18,420 --> 00:05:22,520 You might know that the square root of 225 is 15. 105 00:05:22,519 --> 00:05:25,180 And then from that, you might be able to figure out the 106 00:05:25,180 --> 00:05:29,180 square root of 2.25 is 1.5. 107 00:05:29,180 --> 00:05:32,970 Let me just use the calculator to verify that for you. 108 00:05:32,970 --> 00:05:35,410 So 2.25, take the square root. 109 00:05:35,410 --> 00:05:36,600 It's 1.5. 110 00:05:36,600 --> 00:05:39,170 The principal root is 1.5. 111 00:05:39,170 --> 00:05:40,770 Another square root is negative 1.5. 112 00:05:40,769 --> 00:05:42,019 So it's 1.5. 113 00:05:42,019 --> 00:05:44,159 114 00:05:44,160 --> 00:05:45,980 And then, according to this, this should be equal to 2 115 00:05:45,980 --> 00:05:50,990 times 2.25 is 4.5 minus 6. 116 00:05:50,990 --> 00:05:52,970 Now, is this true? 117 00:05:52,970 --> 00:05:57,350 This is telling us that 1.5 is equal to negative 1.5. 118 00:05:57,350 --> 00:05:59,200 This is not true. 119 00:05:59,199 --> 00:06:03,789 2.5 did not work for this radical equation. 120 00:06:03,790 --> 00:06:06,580 We call this an extraneous solution. 121 00:06:06,579 --> 00:06:13,729 So 2.25 is an extraneous solution. 122 00:06:13,730 --> 00:06:18,069 Now, here's the conundrum: Why did we get 2.25 as an answer? 123 00:06:18,069 --> 00:06:20,310 It looks like we did very valid things the whole way 124 00:06:20,310 --> 00:06:23,829 down, and we got a quadratic, and we got 2.25. 125 00:06:23,829 --> 00:06:25,259 And there's a hint here. 126 00:06:25,259 --> 00:06:28,009 When we substitute 2.25, we get 1.5 is 127 00:06:28,009 --> 00:06:30,550 equal to negative 1.5. 128 00:06:30,550 --> 00:06:34,120 So there's something here, something we did gave us this 129 00:06:34,120 --> 00:06:37,430 solution that doesn't quite apply over here. 130 00:06:37,430 --> 00:06:38,720 And I'll give you another hint. 131 00:06:38,720 --> 00:06:40,090 Let's try it at this step. 132 00:06:40,089 --> 00:06:42,319 If you look at this step, you're going to see that both 133 00:06:42,319 --> 00:06:44,300 solutions actually work. 134 00:06:44,300 --> 00:06:45,990 So you could try it out if you like. 135 00:06:45,990 --> 00:06:47,740 Actually, try it out on your own time. 136 00:06:47,740 --> 00:06:49,865 Put in 2.25 for x here. 137 00:06:49,865 --> 00:06:51,519 You're going to see that it works. 138 00:06:51,519 --> 00:06:55,060 Put in 4 for x here and you see that they both work here. 139 00:06:55,060 --> 00:06:57,360 So they're both valid solutions to that. 140 00:06:57,360 --> 00:07:00,400 141 00:07:00,399 --> 00:07:05,329 So something happened when we squared that made the equation 142 00:07:05,329 --> 00:07:07,069 a little bit different. 143 00:07:07,069 --> 00:07:10,000 There's something slightly different about this equation 144 00:07:10,000 --> 00:07:11,550 than that equation. 145 00:07:11,550 --> 00:07:15,710 And the answer is there's two ways you could think about it. 146 00:07:15,709 --> 00:07:20,430 To go back from this equation to that equation, we take the 147 00:07:20,430 --> 00:07:21,439 square root. 148 00:07:21,439 --> 00:07:24,259 But to be more particular about it, we are taking the 149 00:07:24,259 --> 00:07:27,019 principal root of both sides. 150 00:07:27,019 --> 00:07:30,409 Now, you could take the negative square root as well. 151 00:07:30,410 --> 00:07:33,710 Notice, this is only taking the principal square root. 152 00:07:33,709 --> 00:07:36,430 Going from this right here-- let me be very clear. 153 00:07:36,430 --> 00:07:40,930 This statement, we already established that both of these 154 00:07:40,930 --> 00:07:43,610 solutions, both the valid solution and the extraneous 155 00:07:43,610 --> 00:07:45,350 solution to this radical equation, 156 00:07:45,350 --> 00:07:47,060 satisfy this right here. 157 00:07:47,060 --> 00:07:49,970 Only the valid one satisfies the original problem. 158 00:07:49,970 --> 00:07:52,390 So let me write the equation that both of them satisfy. 159 00:07:52,389 --> 00:07:55,550 Because this is really an interesting conundrum. 160 00:07:55,550 --> 00:07:58,370 And I think it gives you a little bit of a nuance and 161 00:07:58,370 --> 00:07:59,860 kind of tells you what's happening when we take 162 00:07:59,860 --> 00:08:01,319 principal roots of things. 163 00:08:01,319 --> 00:08:04,230 And why when you square both sides, you are, to some 164 00:08:04,230 --> 00:08:06,660 degree, you could either think of it as losing or gaining 165 00:08:06,660 --> 00:08:07,960 some information. 166 00:08:07,959 --> 00:08:11,719 Now, this could be written as x is equal 167 00:08:11,720 --> 00:08:16,450 to 2x minus 6 squared. 168 00:08:16,449 --> 00:08:19,089 This is one valid interpretation of this 169 00:08:19,089 --> 00:08:20,469 equation right here. 170 00:08:20,470 --> 00:08:24,290 But there's a completely other legitimate interpretation of 171 00:08:24,290 --> 00:08:27,220 this equation. 172 00:08:27,220 --> 00:08:31,300 This could also be x is equal to negative 1 173 00:08:31,300 --> 00:08:35,399 times 2x minus 6 squared. 174 00:08:35,399 --> 00:08:37,870 And why are these equal interpretations? 175 00:08:37,870 --> 00:08:41,009 Because when you square the negative 1, the negative 1 176 00:08:41,009 --> 00:08:42,610 will disappear. 177 00:08:42,610 --> 00:08:44,419 These are equivalent statements. 178 00:08:44,419 --> 00:08:47,500 And another way of writing this one, another way of 179 00:08:47,500 --> 00:08:50,629 writing this right here, is that x is equal to-- you 180 00:08:50,629 --> 00:08:52,450 multiply negative 1 times that. 181 00:08:52,450 --> 00:08:58,080 You get negative 2x plus 6 or 6 minus 2x squared. 182 00:08:58,080 --> 00:09:04,560 This and this are two ways of writing that. 183 00:09:04,559 --> 00:09:07,759 Now, when we took our square root or when we-- I guess 184 00:09:07,759 --> 00:09:09,700 there's two ways you can think about it. 185 00:09:09,700 --> 00:09:13,170 When we squared it, we're assuming that this was the 186 00:09:13,169 --> 00:09:15,799 only interpretation, but this was the other one. 187 00:09:15,799 --> 00:09:21,459 So we found two solutions to this, but only 4 satisfies 188 00:09:21,460 --> 00:09:24,060 this interpretation right here. 189 00:09:24,059 --> 00:09:26,799 I hope you get what I'm saying because we're kind of only 190 00:09:26,799 --> 00:09:29,709 taking-- you can imagine the positive square root. 191 00:09:29,710 --> 00:09:32,300 We're not considering the negative square root of this, 192 00:09:32,299 --> 00:09:34,079 because when you take the square root of both sides to 193 00:09:34,080 --> 00:09:36,009 get here, we're only taking the principal root. 194 00:09:36,009 --> 00:09:37,879 Another way to view it-- let me rewrite 195 00:09:37,879 --> 00:09:39,129 the original equation. 196 00:09:39,129 --> 00:09:42,559 197 00:09:42,559 --> 00:09:48,629 We had the square root of x is equal to 2x minus 6. 198 00:09:48,629 --> 00:09:50,620 Now, we said 4 is a solution. 199 00:09:50,620 --> 00:09:52,789 2.25 isn't a solution. 200 00:09:52,789 --> 00:09:56,689 2.25 would've been a solution if we said both of the square 201 00:09:56,690 --> 00:09:59,800 roots of x is equal to 2x minus 6. 202 00:09:59,799 --> 00:10:04,219 Now you try it out and 2.25 will have a 203 00:10:04,220 --> 00:10:05,330 valid solution here. 204 00:10:05,330 --> 00:10:10,050 If you take the negative square root of 2.25, that is 205 00:10:10,049 --> 00:10:15,589 equal to 2 times 2.25, so that is equal to 4.5 minus 6, which 206 00:10:15,590 --> 00:10:17,230 is negative 1.5. 207 00:10:17,230 --> 00:10:18,430 That is true. 208 00:10:18,429 --> 00:10:21,199 The positive version is where you get x is equal to 4. 209 00:10:21,200 --> 00:10:23,250 So that's why we got two solutions. 210 00:10:23,250 --> 00:10:25,899 And if you square this-- maybe this is an easier way to 211 00:10:25,899 --> 00:10:26,299 remember it. 212 00:10:26,299 --> 00:10:31,779 If you square this, you actually get this equation 213 00:10:31,779 --> 00:10:34,149 that both solutions are valid. 214 00:10:34,149 --> 00:10:35,899 Now, you might have found that a little bit 215 00:10:35,899 --> 00:10:37,069 confusing and all of that. 216 00:10:37,070 --> 00:10:38,990 My intention is not to confuse you. 217 00:10:38,990 --> 00:10:41,490 The simple thing to think about when you are solving 218 00:10:41,490 --> 00:10:45,060 radical equations is, look, isolate radicals, square, keep 219 00:10:45,059 --> 00:10:45,739 on solving. 220 00:10:45,740 --> 00:10:47,460 You might get more than one answer. 221 00:10:47,460 --> 00:10:49,180 Plug your answers back in. 222 00:10:49,179 --> 00:10:52,169 Answers that don't work, they're extraneous solutions. 223 00:10:52,169 --> 00:10:54,259 But most of my explanation in this video is really why does 224 00:10:54,259 --> 00:10:56,289 that extraneous solution pop up? 225 00:10:56,289 --> 00:11:00,899 And hopefully, I gave you some intuition that our equation is 226 00:11:00,899 --> 00:11:02,610 the square root of x. 227 00:11:02,610 --> 00:11:05,139 The extraneous solution would be valid if we took the plus 228 00:11:05,139 --> 00:11:09,009 or minus square root of x, not just the principal root. 229 00:11:09,009 --> 00:11:09,133