1 00:00:00,000 --> 00:00:00,490 2 00:00:00,490 --> 00:00:03,190 In this video, I'm going to do several examples of quadratic 3 00:00:03,189 --> 00:00:06,820 equations that are really of a special form, and it's really 4 00:00:06,820 --> 00:00:10,150 a bit of warm-up for the next video that we're going to do 5 00:00:10,150 --> 00:00:11,669 on completing the square. 6 00:00:11,669 --> 00:00:13,910 So let me show you what I'm talking about. 7 00:00:13,910 --> 00:00:20,960 So let's say I have 4x plus 1 squared, minus 8 00:00:20,960 --> 00:00:24,330 8 is equal to 0. 9 00:00:24,329 --> 00:00:26,739 Now, based on everything we've done so far, you might be 10 00:00:26,739 --> 00:00:29,979 tempted to multiply this out, then subtract 8 from the 11 00:00:29,980 --> 00:00:33,380 constant you get out here, and then try to factor it. 12 00:00:33,380 --> 00:00:37,600 And then you're going to have x minus something, times x 13 00:00:37,600 --> 00:00:43,219 minus something else is equal to 0. 14 00:00:43,219 --> 00:00:45,299 And you're going to say, oh, one of these must be equal to 15 00:00:45,299 --> 00:00:47,989 0, so x could be that or that. 16 00:00:47,990 --> 00:00:50,760 We're not going to do that this time, because you might 17 00:00:50,759 --> 00:00:52,809 see something interesting here. 18 00:00:52,810 --> 00:00:56,090 We can solve this without factoring it. 19 00:00:56,090 --> 00:00:57,050 And how do we do that? 20 00:00:57,049 --> 00:00:59,949 Well, what happens if we add 8 to both 21 00:00:59,950 --> 00:01:02,070 sides of this equation? 22 00:01:02,070 --> 00:01:06,480 Then the left-hand side of the equation becomes 4x plus 1 23 00:01:06,480 --> 00:01:09,240 squared, and these 8's cancel out. 24 00:01:09,239 --> 00:01:13,129 The right-hand becomes just a positive 8. 25 00:01:13,129 --> 00:01:15,780 Now, what can we do to both sides of this equation? 26 00:01:15,780 --> 00:01:18,090 And this is just kind of straight, vanilla 27 00:01:18,090 --> 00:01:19,170 equation-solving. 28 00:01:19,170 --> 00:01:21,930 This isn't any kind of fancy factoring. 29 00:01:21,930 --> 00:01:28,700 We can take the square root of both sides of this equation. 30 00:01:28,700 --> 00:01:29,600 We could take the square root. 31 00:01:29,599 --> 00:01:34,409 So 4x plus 1-- I'm just taking the square root of both sides. 32 00:01:34,409 --> 00:01:36,840 You take the square root of both sides, and, of course, 33 00:01:36,840 --> 00:01:39,740 you want to take the positive and the negative square root, 34 00:01:39,739 --> 00:01:43,390 because 4x plus 1 could be the positive square root of 8, or 35 00:01:43,390 --> 00:01:45,400 it could be the negative square root of 8. 36 00:01:45,400 --> 00:01:50,100 So 4x plus 1 is equal to the positive or negative 37 00:01:50,099 --> 00:01:51,859 square root of 8. 38 00:01:51,859 --> 00:01:55,450 Instead of 8, let me write 8 as 4 times 2. 39 00:01:55,450 --> 00:01:58,350 We all know that's what 8 is, and obviously the square root 40 00:01:58,349 --> 00:02:02,819 of 4x plus 1 squared is 4x plus 1. 41 00:02:02,819 --> 00:02:09,500 So we get 4x plus 1 is equal to-- we can factor out the 4, 42 00:02:09,500 --> 00:02:12,840 or the square root of 4, which is 2-- is equal to the plus or 43 00:02:12,840 --> 00:02:17,250 minus times 2 times the square root of 2, right? 44 00:02:17,250 --> 00:02:19,460 Square root of 4 times square root of 2 is the same thing as 45 00:02:19,460 --> 00:02:23,290 square root of 4 times the square root of 2, plus or 46 00:02:23,289 --> 00:02:26,849 minus the square root of 4 is that 2 right there. 47 00:02:26,849 --> 00:02:29,969 Now, it might look like a really bizarro equation, with 48 00:02:29,969 --> 00:02:32,340 this plus or minus 2 times the square of 2, 49 00:02:32,340 --> 00:02:34,020 but it really isn't. 50 00:02:34,020 --> 00:02:36,010 These are actually two numbers here, and we're actually 51 00:02:36,009 --> 00:02:38,209 simultaneously solving two equations. 52 00:02:38,210 --> 00:02:43,980 We could write this as 4x plus 1 is equal to the positive 2, 53 00:02:43,979 --> 00:02:52,359 square root of 2, or 4x plus 1 is equal to negative 2 times 54 00:02:52,360 --> 00:02:53,510 the square root of 2. 55 00:02:53,509 --> 00:02:56,109 This one statement is equivalent to this right here, 56 00:02:56,110 --> 00:02:59,230 because we have this plus or minus here, this or statement. 57 00:02:59,229 --> 00:03:01,709 Let me solve all of these simultaneously. 58 00:03:01,710 --> 00:03:05,360 So if I subtract 1 from both sides of this 59 00:03:05,360 --> 00:03:07,030 equation, what do I have? 60 00:03:07,030 --> 00:03:10,330 On the left-hand side, I'm just left with 4x. 61 00:03:10,330 --> 00:03:13,330 On the right-hand side, I have-- you can't really 62 00:03:13,330 --> 00:03:15,260 mathematically, I mean, you could do them if you had a 63 00:03:15,259 --> 00:03:19,209 calculator, but I'll just leave it as negative 1 plus or 64 00:03:19,210 --> 00:03:23,570 minus the square root, or 2 times the square root of 2. 65 00:03:23,569 --> 00:03:25,199 That's what 4x is equal to. 66 00:03:25,199 --> 00:03:28,560 If we did it here, as two separate equations, same idea. 67 00:03:28,560 --> 00:03:32,715 Subtract 1 from both sides of this equation, you get 4x is 68 00:03:32,715 --> 00:03:36,310 equal to negative 1 plus 2, times the square root of 2. 69 00:03:36,310 --> 00:03:39,150 This equation, subtract 1 from both sides. 70 00:03:39,150 --> 00:03:44,090 4x is equal to negative 1 minus 2, times the 71 00:03:44,090 --> 00:03:45,560 square root of 2. 72 00:03:45,560 --> 00:03:52,969 This statement right here is completely equivalent to these 73 00:03:52,969 --> 00:03:53,879 two statements. 74 00:03:53,879 --> 00:03:57,590 Now, last step, we just have to divide both sides by 4, so 75 00:03:57,590 --> 00:04:02,770 you divide both sides by 4, and you get x is equal to 76 00:04:02,770 --> 00:04:06,620 negative 1 plus or minus 2, times the square 77 00:04:06,620 --> 00:04:09,569 root of 2, over 4. 78 00:04:09,569 --> 00:04:11,669 Now, this statement is completely equivalent to 79 00:04:11,669 --> 00:04:16,550 dividing each of these by 4, and you get x is equal to 80 00:04:16,550 --> 00:04:21,470 negative 1 plus 2, times the square root 2, over 4. 81 00:04:21,470 --> 00:04:22,890 This is one solution. 82 00:04:22,889 --> 00:04:27,269 And then the other solution is x is equal to negative 1 minus 83 00:04:27,269 --> 00:04:31,370 2 roots of 2, all of that over 4. 84 00:04:31,370 --> 00:04:35,340 That statement and these two statements are equivalent. 85 00:04:35,339 --> 00:04:38,229 And if you want, I encourage you to-- let's substitute one 86 00:04:38,230 --> 00:04:41,280 of these back in, just so you feel confident that something 87 00:04:41,279 --> 00:04:45,959 as bizarro as one of these expressions can be a solution 88 00:04:45,959 --> 00:04:48,969 to a nice, vanilla-looking equation like this. 89 00:04:48,970 --> 00:04:50,870 So let's substitute it back in. 90 00:04:50,870 --> 00:04:58,290 4 times x, or 4 times negative 1, plus 2 root 2, over 4, plus 91 00:04:58,290 --> 00:05:03,450 1 squared, minus 8 is equal to 0. 92 00:05:03,449 --> 00:05:07,349 Now, these 4's cancel out, so you're left with negative 1 93 00:05:07,350 --> 00:05:12,260 plus 2 roots 2, plus 1, squared, minus 94 00:05:12,259 --> 00:05:14,409 8 is equal to 0. 95 00:05:14,410 --> 00:05:17,410 This negative 1 and this positive 1 cancel out, so 96 00:05:17,410 --> 00:05:21,689 you're left with 2 roots of 2 squared, minus 97 00:05:21,689 --> 00:05:24,839 8 is equal to 0. 98 00:05:24,839 --> 00:05:28,679 And then what are you going to have here? 99 00:05:28,680 --> 00:05:33,459 So when you square this, you get 4 times 2, minus 8 is 100 00:05:33,459 --> 00:05:38,060 equal to 0, which is true. 101 00:05:38,060 --> 00:05:40,300 8 minus 8 is equal to 0. 102 00:05:40,300 --> 00:05:42,740 And if you try this one out, you're going to get the exact 103 00:05:42,740 --> 00:05:44,180 same answer. 104 00:05:44,180 --> 00:05:45,220 Let's do another one like this. 105 00:05:45,220 --> 00:05:46,960 And remember, these are special forms where we have 106 00:05:46,959 --> 00:05:51,469 squares of binomials in our expression. 107 00:05:51,470 --> 00:05:53,620 And we're going to see that the entire quadratic formula 108 00:05:53,620 --> 00:05:55,870 is actually derived from a notion like this, because you 109 00:05:55,870 --> 00:06:00,250 can actually turn any, you can turn any, quadratic equation 110 00:06:00,250 --> 00:06:05,050 into a perfect square equalling something else. 111 00:06:05,050 --> 00:06:06,525 We'll see that two videos from now. 112 00:06:06,524 --> 00:06:08,259 But let's get a little warmed up just seeing 113 00:06:08,259 --> 00:06:09,409 this type of thing. 114 00:06:09,410 --> 00:06:16,650 So let's say you have x squared minus 10x, plus 25 is 115 00:06:16,649 --> 00:06:18,679 equal to 9. 116 00:06:18,680 --> 00:06:20,920 Now, once again your temptation-- and it's not a 117 00:06:20,920 --> 00:06:23,960 bad temptation-- would be to subtract 9 from both sides, so 118 00:06:23,959 --> 00:06:27,319 you get a 0 on the right-hand side, but before you do that, 119 00:06:27,319 --> 00:06:31,050 just inspect this really fast. And say, hey, is this just 120 00:06:31,050 --> 00:06:34,139 maybe a perfect square of a binomial? 121 00:06:34,139 --> 00:06:38,149 And you see-- well, what two numbers when I multiply them I 122 00:06:38,149 --> 00:06:41,759 get positive 25, and when I add them I get negative 10? 123 00:06:41,759 --> 00:06:44,379 And hopefully negative 5 jumps out at you. 124 00:06:44,379 --> 00:06:52,060 So this expression right here is x minus 5, times x minus 5. 125 00:06:52,060 --> 00:06:56,495 So this left-hand side can be written as x minus 5 squared, 126 00:06:56,495 --> 00:06:59,910 and the right-hand side is still 9. 127 00:06:59,910 --> 00:07:01,540 And I want to really emphasize. 128 00:07:01,540 --> 00:07:03,860 I don't want this to ruin all of the training you've gotten 129 00:07:03,860 --> 00:07:05,280 on factoring so far. 130 00:07:05,279 --> 00:07:08,259 We can only do this when this is a perfect square. 131 00:07:08,259 --> 00:07:12,209 If you got, like, x minus 3, times x plus 4, and that would 132 00:07:12,209 --> 00:07:15,120 be equal to 9, that would be a dead end. 133 00:07:15,120 --> 00:07:16,579 You wouldn't be able to really do anything 134 00:07:16,579 --> 00:07:17,490 constructive with that. 135 00:07:17,490 --> 00:07:21,079 Only because this is a perfect square, can we now say x minus 136 00:07:21,079 --> 00:07:23,639 5 squared is equal to 9, and now we can take the square 137 00:07:23,639 --> 00:07:25,060 root of both sides. 138 00:07:25,060 --> 00:07:31,280 So we could say that x minus 5 is equal to plus or minus 3. 139 00:07:31,279 --> 00:07:36,049 Add 5 to both sides of this equation, you get x is equal 140 00:07:36,050 --> 00:07:43,050 to 5 plus or minus 3, or x is equal to-- what's 5 plus 3? 141 00:07:43,050 --> 00:07:49,780 Well, x could be 8 or x could be equal to 5 minus 3, or x is 142 00:07:49,779 --> 00:07:51,339 equal to 2. 143 00:07:51,339 --> 00:07:54,889 Now, we could have done this equation, this quadratic 144 00:07:54,889 --> 00:07:58,069 equation, the traditional way, the way you were 145 00:07:58,069 --> 00:07:59,159 tempted to do it. 146 00:07:59,160 --> 00:08:01,600 What happens if you subtract 9 from both 147 00:08:01,600 --> 00:08:03,230 sides of this equation? 148 00:08:03,230 --> 00:08:07,650 You'll get x squared minus 10x. 149 00:08:07,649 --> 00:08:09,819 And what's 25 minus 9? 150 00:08:09,819 --> 00:08:16,040 25 minus 9 is 16, and that would be equal to 0. 151 00:08:16,040 --> 00:08:18,970 And here, this would be just a traditional factoring problem, 152 00:08:18,970 --> 00:08:21,400 the type that we've seen in the last few videos. 153 00:08:21,399 --> 00:08:23,709 What two numbers, when you take their product, you get 154 00:08:23,709 --> 00:08:29,019 positive 16, and when you sum them you get negative 10? 155 00:08:29,019 --> 00:08:31,799 And maybe negative 8 and negative 2 156 00:08:31,800 --> 00:08:33,389 jump into your brain. 157 00:08:33,389 --> 00:08:38,953 So we get x minus 8, times x minus 2 is equal to 0. 158 00:08:38,953 --> 00:08:43,709 And so x could be equal to 8 or x could be equal to 2. 159 00:08:43,710 --> 00:08:46,450 That's the fun thing about algebra: you can do things in 160 00:08:46,450 --> 00:08:49,670 two completely different ways, but as long as you do them in 161 00:08:49,669 --> 00:08:52,209 algebraically-valid ways, you're not going to get 162 00:08:52,210 --> 00:08:55,170 different answers. 163 00:08:55,169 --> 00:08:57,329 And on some level, if you recognize this, this is easier 164 00:08:57,330 --> 00:08:58,950 because you didn't have to do that little game in your head, 165 00:08:58,950 --> 00:09:01,129 in terms of, oh, what two numbers, when you multiply 166 00:09:01,129 --> 00:09:02,840 them you get 16, and when you add them you get negative 10? 167 00:09:02,840 --> 00:09:05,340 Here, you just said, OK, this is x minus 5-- oh, I guess you 168 00:09:05,340 --> 00:09:06,019 did have to do it. 169 00:09:06,019 --> 00:09:10,019 You had to say, oh, 5 times 5 is 25, and negative 10 is 170 00:09:10,019 --> 00:09:11,159 negative 5 plus negative 5. 171 00:09:11,159 --> 00:09:14,049 So I take that back, you still have to do that little 172 00:09:14,049 --> 00:09:15,490 game in your head. 173 00:09:15,490 --> 00:09:18,700 So let's do another one. 174 00:09:18,700 --> 00:09:22,879 Let's do one more of these, just to really get ourselves 175 00:09:22,879 --> 00:09:25,230 nice and warmed up here. 176 00:09:25,230 --> 00:09:38,139 So, let's say we have x squared plus 18x, plus 81 is 177 00:09:38,139 --> 00:09:41,470 equal to 1. 178 00:09:41,470 --> 00:09:42,940 So once again, we can do it in two ways. 179 00:09:42,940 --> 00:09:46,580 We could subtract 1 from both sides, or we could recognize 180 00:09:46,580 --> 00:09:53,139 that this is x plus 9, times x plus 9. 181 00:09:53,139 --> 00:09:58,009 This right here, 9 times 9 is 81, 9 plus 9 is 18. 182 00:09:58,009 --> 00:10:01,210 So we can write our equation as x plus 9 183 00:10:01,210 --> 00:10:03,759 squared is equal to 1. 184 00:10:03,759 --> 00:10:06,939 Take the square root of both sides, you get x plus 9 is 185 00:10:06,940 --> 00:10:09,240 equal to plus or minus the square root of 1, 186 00:10:09,240 --> 00:10:10,909 which is just 1. 187 00:10:10,909 --> 00:10:15,139 So x is equal to-- subtract 9 from both sides-- negative 9 188 00:10:15,139 --> 00:10:16,865 plus or minus 1. 189 00:10:16,865 --> 00:10:21,059 And that means that x could be equal to-- negative 9 plus 1 190 00:10:21,059 --> 00:10:26,619 is negative 8, or x could be equal to-- negative 9 minus 1, 191 00:10:26,620 --> 00:10:28,929 which is negative 10. 192 00:10:28,929 --> 00:10:31,559 And once again, you could have done this the traditional way. 193 00:10:31,559 --> 00:10:33,729 You could have subtracted 1 from both sides and you would 194 00:10:33,730 --> 00:10:41,240 have gotten x squared plus 18x, plus 80 is equal to 0. 195 00:10:41,240 --> 00:10:44,370 And you'd say, hey, gee, 8 times 10 is 80, 8 plus 10 is 196 00:10:44,370 --> 00:10:50,850 18, so you get x plus 8, times x plus 10 is equal to 0. 197 00:10:50,850 --> 00:10:54,639 And then you'd get x could be equal to negative 8, or x 198 00:10:54,639 --> 00:10:57,360 could be equal to negative 10. 199 00:10:57,360 --> 00:10:58,710 That was good warm up. 200 00:10:58,710 --> 00:11:02,250 Now, I think we're ready to tackle completing the square. 201 00:11:02,250 --> 00:11:02,466