1 00:00:00,000 --> 00:00:00,540 2 00:00:00,540 --> 00:00:03,810 Complete the square on the general quadratic equation. 3 00:00:03,810 --> 00:00:08,949 We have ax squared plus bx plus c is equal to 0. 4 00:00:08,949 --> 00:00:11,160 So whenever I complete the square, actually whenever I 5 00:00:11,160 --> 00:00:13,620 deal with any of these types of quadratic equations, I 6 00:00:13,619 --> 00:00:18,710 always like to not have an a, or a non 1 coefficient, on the 7 00:00:18,710 --> 00:00:21,630 x squared terms. So let's make it into a 1 coefficient. 8 00:00:21,629 --> 00:00:23,140 And the easiest way to do that is just divide 9 00:00:23,140 --> 00:00:24,330 everything by a. 10 00:00:24,329 --> 00:00:27,929 So we divide every term on the left side by a, and of course 11 00:00:27,929 --> 00:00:31,109 we have to divide the right side by a as well. 12 00:00:31,109 --> 00:00:34,299 And so the left side will become x 13 00:00:34,299 --> 00:00:36,890 squared plus b over ax. 14 00:00:36,890 --> 00:00:39,500 15 00:00:39,500 --> 00:00:41,630 And then I'll write c over a over here. 16 00:00:41,630 --> 00:00:43,969 So we have some room to add and subtract things so we can 17 00:00:43,969 --> 00:00:45,619 really complete the square. 18 00:00:45,619 --> 00:00:51,769 So plus c over a is equal to 0 divided by a, which is just 19 00:00:51,770 --> 00:00:54,960 going to be equal to 0. 20 00:00:54,960 --> 00:00:57,799 Now when we complete the square, we've seen this 21 00:00:57,799 --> 00:01:01,709 multiple times before, what we want to do is take the 22 00:01:01,710 --> 00:01:04,819 coefficient on the x term right here, it's b over a, 23 00:01:04,819 --> 00:01:05,750 take half of it. 24 00:01:05,750 --> 00:01:10,980 This right here is two times b over 2a. 25 00:01:10,980 --> 00:01:11,420 Right? 26 00:01:11,420 --> 00:01:12,420 The 2's cancel out. 27 00:01:12,420 --> 00:01:14,909 This is just 2 times b over 2a. 28 00:01:14,909 --> 00:01:19,019 So you take half of it, half of b over a is b over 2a. 29 00:01:19,019 --> 00:01:22,269 You take half of it, and then you square it, and you add it 30 00:01:22,269 --> 00:01:22,829 right here. 31 00:01:22,829 --> 00:01:26,564 So plus b squared-- Let me write it this way. 32 00:01:26,564 --> 00:01:31,780 b over 2a squared. 33 00:01:31,780 --> 00:01:34,799 And of course I can't just add something to one side of the 34 00:01:34,799 --> 00:01:36,509 equation, that would change the equation. 35 00:01:36,510 --> 00:01:38,719 I also either have to add it to the other side, or just 36 00:01:38,719 --> 00:01:41,140 subtract it from the same side that I'm adding it to. 37 00:01:41,140 --> 00:01:46,120 So I'll also subtract the b over 2a 38 00:01:46,120 --> 00:01:48,410 squared just like that. 39 00:01:48,409 --> 00:01:52,769 Now, the whole point of doing this is so these first three 40 00:01:52,769 --> 00:01:57,199 terms right here are a perfect square trinomial. 41 00:01:57,200 --> 00:01:59,290 That's what completing the square is all about. 42 00:01:59,290 --> 00:02:01,150 And we've seen the pattern multiple times. 43 00:02:01,150 --> 00:02:08,590 If I have, let's say, m plus n squared-- and I'm using m and 44 00:02:08,590 --> 00:02:10,650 n so we don't get confused with the a's and b's 45 00:02:10,650 --> 00:02:11,860 and x's over here. 46 00:02:11,860 --> 00:02:14,430 But if I have m plus n squared, we've seen multiple 47 00:02:14,430 --> 00:02:20,250 times, that's going to be equal to m squared plus 2mn 48 00:02:20,250 --> 00:02:22,030 plus n squared. 49 00:02:22,030 --> 00:02:23,759 And here we have that pattern now. 50 00:02:23,759 --> 00:02:25,769 That's the whole point behind completing the square. 51 00:02:25,770 --> 00:02:29,390 That's the whole point behind taking half of b over a-- 52 00:02:29,389 --> 00:02:31,779 that's b over 2a-- and then squaring it and adding it 53 00:02:31,780 --> 00:02:32,140 right here. 54 00:02:32,139 --> 00:02:33,479 We now fit that pattern. 55 00:02:33,479 --> 00:02:35,539 m is x. 56 00:02:35,539 --> 00:02:37,389 n is b over 2a. 57 00:02:37,389 --> 00:02:42,139 And 2mn, if I take an x times a b over 2a, and multiply that 58 00:02:42,139 --> 00:02:44,419 by 2, I get b over ax. 59 00:02:44,419 --> 00:02:46,780 So this expression, right here, this trinomial, the 60 00:02:46,780 --> 00:02:50,250 first three terms, it is a perfect square trinomial, and 61 00:02:50,250 --> 00:02:58,409 we can write it as x plus b over 2a squared. 62 00:02:58,409 --> 00:02:59,840 And then of course we have all this other 63 00:02:59,840 --> 00:03:01,450 business right here. 64 00:03:01,449 --> 00:03:04,539 And then of course we have all of this 65 00:03:04,539 --> 00:03:05,959 other stuff right here. 66 00:03:05,960 --> 00:03:11,360 Which is negative b over, and let me just actually 67 00:03:11,360 --> 00:03:12,320 square it for you. 68 00:03:12,319 --> 00:03:17,409 So b over 2a squared is negative b 69 00:03:17,409 --> 00:03:20,969 squared over 4a squared. 70 00:03:20,969 --> 00:03:23,659 And then I have this plus c over a. 71 00:03:23,659 --> 00:03:25,979 But let's write it with the same denominator here. 72 00:03:25,979 --> 00:03:31,530 So I could have c over a, or I could multiply the numerator 73 00:03:31,530 --> 00:03:34,280 and the denominator by 4a. 74 00:03:34,280 --> 00:03:37,000 So if I multiply the numerator by 4a, I get 4ac. 75 00:03:37,000 --> 00:03:43,039 If I multiply the denominator by 4a, I get 4a squared. 76 00:03:43,039 --> 00:03:44,289 And the whole reason why I multiplied the numerator and 77 00:03:44,289 --> 00:03:48,060 the denominator by 4a was so that we have the same 78 00:03:48,060 --> 00:03:49,400 denominator right here. 79 00:03:49,400 --> 00:03:52,939 And, of course, that is going to be equal to 0. 80 00:03:52,939 --> 00:03:55,539 And we could simplify it a little bit more, or actually, 81 00:03:55,539 --> 00:03:57,549 well yeah we'll just simplify it next in the next step. 82 00:03:57,550 --> 00:03:59,580 We don't want to skip too many steps here. 83 00:03:59,580 --> 00:04:05,219 So you have x plus b over 2a squared. 84 00:04:05,219 --> 00:04:09,379 And then we could say, plus-- we could put the 4ac first, so 85 00:04:09,379 --> 00:04:12,509 we could say-- actually let's just say, plus negative b 86 00:04:12,509 --> 00:04:23,230 squared, plus 4ac, all of that over 4a squared is equal to 0. 87 00:04:23,230 --> 00:04:25,650 I didn't put the 4ac first, I just put the negative b 88 00:04:25,649 --> 00:04:26,279 squared there. 89 00:04:26,279 --> 00:04:30,879 Now let's isolate this squared binomial on 90 00:04:30,879 --> 00:04:31,600 the left hand side. 91 00:04:31,600 --> 00:04:34,600 And the easiest way we can do that is to subtract this thing 92 00:04:34,600 --> 00:04:36,150 from both sides of the equation. 93 00:04:36,149 --> 00:04:38,039 So let's do that. 94 00:04:38,040 --> 00:04:42,540 So you can imagine if we add b-- let me do this in a 95 00:04:42,540 --> 00:04:44,060 different color. 96 00:04:44,060 --> 00:04:51,540 If we were to add positive b squared minus 4ac over 4a 97 00:04:51,540 --> 00:04:53,950 squared on the left hand side, those will cancel out and 98 00:04:53,949 --> 00:04:55,569 we're also going to add it on the right hand side. 99 00:04:55,569 --> 00:05:01,779 Positive b squared minus 4ac over 4a squared. 100 00:05:01,779 --> 00:05:03,649 Anything I do the left I have to do the right. 101 00:05:03,649 --> 00:05:04,899 What do we end up with? 102 00:05:04,899 --> 00:05:07,539 103 00:05:07,540 --> 00:05:10,970 We end up with, on the left hand side, these two guys 104 00:05:10,970 --> 00:05:11,470 cancel out. 105 00:05:11,470 --> 00:05:13,460 We have the same denomenator, when you add the numerators, 106 00:05:13,459 --> 00:05:14,799 that cancels with that. 107 00:05:14,800 --> 00:05:17,270 The 4ac cancels with the negative 4ac, these just 108 00:05:17,269 --> 00:05:18,639 completely cancel out. 109 00:05:18,639 --> 00:05:22,060 And on the left hand side, you just have x 110 00:05:22,060 --> 00:05:25,639 plus b over 2a squared. 111 00:05:25,639 --> 00:05:30,089 And on the right hand side, you have that being equal to b 112 00:05:30,089 --> 00:05:34,659 squared minus-- let me do that in a blue color. 113 00:05:34,660 --> 00:05:43,189 b squared minus 4ac, all of that over 4a squared. 114 00:05:43,189 --> 00:05:46,329 Now the next thing we probably want to do, if we want to 115 00:05:46,329 --> 00:05:51,310 really solve for x, is to take the square root of both sides 116 00:05:51,310 --> 00:05:52,509 of this equation. 117 00:05:52,509 --> 00:05:53,300 So let's do that. 118 00:05:53,300 --> 00:05:57,050 Let's take the square root of both sides of this equation. 119 00:05:57,050 --> 00:06:02,220 And if, when we do that-- We don't want to only take the 120 00:06:02,220 --> 00:06:06,120 positive square root because x plus b be over 2a could be a 121 00:06:06,120 --> 00:06:08,220 negative number, or it could be a positive number. 122 00:06:08,220 --> 00:06:10,070 So we want to take the positive and 123 00:06:10,069 --> 00:06:11,740 negative square root. 124 00:06:11,740 --> 00:06:17,030 So we could say that the square root, we could put the 125 00:06:17,029 --> 00:06:19,419 positive or negative here or, since we're taking the square 126 00:06:19,420 --> 00:06:21,480 root of both sides, we could put the 127 00:06:21,480 --> 00:06:23,120 positive or negative there. 128 00:06:23,120 --> 00:06:25,800 If you put the positive or negative on both sides, it's 129 00:06:25,800 --> 00:06:27,389 really just telling you the same thing. 130 00:06:27,389 --> 00:06:30,289 It really is all the different combinations. 131 00:06:30,290 --> 00:06:32,720 If the negative square root over here equals the negative 132 00:06:32,720 --> 00:06:36,750 square root over here, then it's just another combination 133 00:06:36,750 --> 00:06:38,279 of the different positives and negatives. 134 00:06:38,279 --> 00:06:42,119 So you could just write it as, this square root is equal to 135 00:06:42,120 --> 00:06:44,410 the plus or minus square root of b squared 136 00:06:44,410 --> 00:06:47,710 minus 4ac over 4a squared. 137 00:06:47,709 --> 00:06:49,709 Now what does this simplify to? 138 00:06:49,709 --> 00:07:01,139 Well, the left hand side just becomes x plus b over 2a is 139 00:07:01,139 --> 00:07:04,099 equal to-- and now it gets interesting. 140 00:07:04,100 --> 00:07:06,960 And you might even start recognizing parts of it. 141 00:07:06,959 --> 00:07:10,149 So let's take the plus or minus square root of the top. 142 00:07:10,149 --> 00:07:12,479 What is that going to be? 143 00:07:12,480 --> 00:07:14,550 And you could just take the plus or minus only of the top 144 00:07:14,550 --> 00:07:16,560 because, once again, the same principles apply. 145 00:07:16,560 --> 00:07:18,370 There's no reason why you have to do a plus or minus over a 146 00:07:18,370 --> 00:07:22,069 plus and minus and a plus or minus on the left hand side. 147 00:07:22,069 --> 00:07:24,939 There's only one combination here, where there's only one 148 00:07:24,939 --> 00:07:25,910 plus or minus on the numerator. 149 00:07:25,910 --> 00:07:28,630 I apologize if that confuses you. 150 00:07:28,629 --> 00:07:34,029 So let's write this as the plus or minus square root of b 151 00:07:34,029 --> 00:07:37,769 squared minus 4ac over-- What's the square 152 00:07:37,769 --> 00:07:40,049 root of 4a a squared? 153 00:07:40,050 --> 00:07:41,300 Well, it's just going to be 2a. 154 00:07:41,300 --> 00:07:43,865 155 00:07:43,865 --> 00:07:44,420 Right? 156 00:07:44,420 --> 00:07:45,930 The square root of 4 is 2. 157 00:07:45,930 --> 00:07:47,800 The square root of a squared is a. 158 00:07:47,800 --> 00:07:49,100 And we're almost there. 159 00:07:49,100 --> 00:07:52,620 To solve for x, we just have to subtract b over 2a from 160 00:07:52,620 --> 00:07:54,069 both sides. 161 00:07:54,069 --> 00:07:57,639 We just have to subtract b over 2a from both sides of 162 00:07:57,639 --> 00:07:58,800 this equation. 163 00:07:58,800 --> 00:08:01,360 The left hand side, we just end up with our x. 164 00:08:01,360 --> 00:08:06,500 And then the right hand side, we have a negative b over 2a 165 00:08:06,500 --> 00:08:13,199 plus or minus the square root of b squared minus 4ac-- all 166 00:08:13,199 --> 00:08:14,629 of that over 2a. 167 00:08:14,629 --> 00:08:16,889 Since we have the same denominator, we can write this 168 00:08:16,889 --> 00:08:22,709 as negative b plus or minus the square root of b squared 169 00:08:22,709 --> 00:08:27,729 minus 4ac-- all of that over 2a. 170 00:08:27,730 --> 00:08:28,259 And we're done. 171 00:08:28,259 --> 00:08:31,300 We've solved for the x's and you see there's actually two 172 00:08:31,300 --> 00:08:32,288 solutions here. 173 00:08:32,288 --> 00:08:34,340 There's one where you take the positive square root, and 174 00:08:34,340 --> 00:08:35,620 there's another solution where you take the 175 00:08:35,620 --> 00:08:36,379 negative square root. 176 00:08:36,379 --> 00:08:39,428 If this square root exists, and if the positive and 177 00:08:39,428 --> 00:08:41,089 negative-- and if it's not 0, you're 178 00:08:41,090 --> 00:08:42,788 going to have two solutions. 179 00:08:42,788 --> 00:08:46,360 And this, right here, this result we have is-- Look, you 180 00:08:46,360 --> 00:08:50,539 give me any quadratic equation, you give me the a, 181 00:08:50,539 --> 00:08:54,089 the b, and the c, we could now substitute it into this 182 00:08:54,090 --> 00:08:57,340 formula essentially we just derived right here, and I'll 183 00:08:57,340 --> 00:09:00,790 give you the roots, I'll give you the x's for that quadratic 184 00:09:00,789 --> 00:09:03,289 equation-- the x's that satisfy 185 00:09:03,289 --> 00:09:04,490 that quadratic equation. 186 00:09:04,490 --> 00:09:07,960 And this formula, right here, for solving any quadratic 187 00:09:07,960 --> 00:09:10,040 equation is called the quadratic formula. 188 00:09:10,039 --> 00:09:13,829 189 00:09:13,830 --> 00:09:15,820 And you could see it just comes straight out of 190 00:09:15,820 --> 00:09:17,010 completing the square. 191 00:09:17,009 --> 00:09:19,569 There's no mystery, magic here. 192 00:09:19,570 --> 00:09:21,700 But it's easily one of the most useful formulas in 193 00:09:21,700 --> 00:09:22,165 mathematics. 194 00:09:22,164 --> 00:09:23,929 And I'm usually not a huge proponent 195 00:09:23,929 --> 00:09:25,069 of memorizing things. 196 00:09:25,070 --> 00:09:28,570 But it probably will benefit you in life if you did. 197 00:09:28,570 --> 00:09:30,390 Hope you enjoyed that.