1 00:00:00,000 --> 00:00:00,750 2 00:00:00,750 --> 00:00:04,809 Use the quadratic formula to solve the equation, 0 is equal 3 00:00:04,809 --> 00:00:10,500 to negative 7q squared plus 2q plus 9. 4 00:00:10,500 --> 00:00:14,230 Now, the quadratic formula, it applies to any quadratic 5 00:00:14,230 --> 00:00:16,750 equation of the form-- we could put the 0 on 6 00:00:16,750 --> 00:00:17,600 the left hand side. 7 00:00:17,600 --> 00:00:24,160 0 is equal to ax squared plus bx plus c. 8 00:00:24,160 --> 00:00:27,170 And we generally deal with x's, in this problem we're 9 00:00:27,170 --> 00:00:28,230 dealing with q's. 10 00:00:28,230 --> 00:00:29,980 But the quadratic formula says, look, if you have a 11 00:00:29,980 --> 00:00:34,149 quadratic equation of this form, that the solutions of 12 00:00:34,149 --> 00:00:37,609 this equation are going to be x is going to be equal to 13 00:00:37,609 --> 00:00:44,329 negative b plus or minus the square root of b squared minus 14 00:00:44,329 --> 00:00:49,149 4ac-- all of that over 2a. 15 00:00:49,149 --> 00:00:51,000 And this is actually two solutions here, because 16 00:00:51,000 --> 00:00:53,350 there's one solution where you take the positive square root 17 00:00:53,350 --> 00:00:54,910 and there's another solution where you take 18 00:00:54,909 --> 00:00:57,029 the negative root. 19 00:00:57,030 --> 00:00:58,679 So it gives you both roots of this. 20 00:00:58,679 --> 00:01:02,030 So if we look at the quadratic equation that we need to solve 21 00:01:02,030 --> 00:01:03,250 here, we can just pattern match. 22 00:01:03,250 --> 00:01:05,980 We're dealing with q's, not x's, but this is the same 23 00:01:05,980 --> 00:01:06,850 general idea. 24 00:01:06,849 --> 00:01:08,419 It could be x's if you like. 25 00:01:08,420 --> 00:01:13,549 And if we look at it, negative 7 corresponds to a. 26 00:01:13,549 --> 00:01:14,599 That is our a. 27 00:01:14,599 --> 00:01:18,319 It's the coefficient on the second degree term. 28 00:01:18,319 --> 00:01:21,289 2 corresponds to b. 29 00:01:21,290 --> 00:01:24,250 It is the coefficient on the first degree term. 30 00:01:24,250 --> 00:01:26,879 And then 9 corresponds to c. 31 00:01:26,879 --> 00:01:29,250 It's the constant. 32 00:01:29,250 --> 00:01:31,439 So, let's just apply the quadratic formula. 33 00:01:31,439 --> 00:01:34,509 The quadratic formula will tell us that the solutions-- 34 00:01:34,510 --> 00:01:38,600 the q's that satisfy this equation-- q will be equal to 35 00:01:38,599 --> 00:01:40,439 negative b. 36 00:01:40,439 --> 00:01:42,390 b is 2. 37 00:01:42,390 --> 00:01:53,840 Plus or minus the square root of b squared, of 2 squared, 38 00:01:53,840 --> 00:02:08,219 minus 4 times a times negative 7 times c, which is 9. 39 00:02:08,219 --> 00:02:10,639 And all of that over 2a. 40 00:02:10,639 --> 00:02:16,020 All of that over 2 times a, which is once 41 00:02:16,020 --> 00:02:17,400 again negative 7. 42 00:02:17,400 --> 00:02:20,256 43 00:02:20,256 --> 00:02:22,759 And then we just have to evaluate this. 44 00:02:22,759 --> 00:02:29,189 So this is going to be equal to negative 2 plus or minus 45 00:02:29,189 --> 00:02:35,159 the square root of-- let's see, 2 squared is 4-- and then 46 00:02:35,159 --> 00:02:37,961 if we just take this part right here, if we just take 47 00:02:37,961 --> 00:02:42,469 the negative 4 times negative 7 times 9, this negative and 48 00:02:42,469 --> 00:02:43,889 that negative is going to cancel out. 49 00:02:43,889 --> 00:02:45,959 So it's just going to become a positive number. 50 00:02:45,960 --> 00:02:49,219 And 4 times 7 times 9. 51 00:02:49,219 --> 00:02:52,719 4 times 9 is 36. 52 00:02:52,719 --> 00:02:54,270 36 times 7. 53 00:02:54,270 --> 00:02:55,710 Let's do it up here. 54 00:02:55,710 --> 00:02:58,060 36 times 7. 55 00:02:58,060 --> 00:03:00,780 7 times 6 is 42. 56 00:03:00,780 --> 00:03:03,349 7 times 3, or 3 times 7 is 21. 57 00:03:03,349 --> 00:03:05,500 Plus 4 is 25. 58 00:03:05,500 --> 00:03:06,419 252. 59 00:03:06,419 --> 00:03:09,289 So this becomes 4 plus 252. 60 00:03:09,289 --> 00:03:11,429 Remember, you have a negative 7 and you 61 00:03:11,430 --> 00:03:12,830 have a minus out front. 62 00:03:12,830 --> 00:03:16,020 Those cancel out, that's why we have a positive 252 for 63 00:03:16,020 --> 00:03:17,950 that part right there. 64 00:03:17,949 --> 00:03:22,219 And then our denominator, 2 times negative 65 00:03:22,219 --> 00:03:24,800 7 is negative 14. 66 00:03:24,800 --> 00:03:26,050 Now what does this equal? 67 00:03:26,050 --> 00:03:28,370 68 00:03:28,370 --> 00:03:33,460 Well, we have this is equal to negative 2 plus or minus the 69 00:03:33,460 --> 00:03:36,080 square root of-- what's 4 plus 252? 70 00:03:36,080 --> 00:03:38,410 It's just 256. 71 00:03:38,409 --> 00:03:40,859 All of that over negative 14. 72 00:03:40,860 --> 00:03:43,600 And what's 256? 73 00:03:43,599 --> 00:03:46,579 What's the square root of 256? 74 00:03:46,580 --> 00:03:47,360 It's 16. 75 00:03:47,360 --> 00:03:48,420 You can try it out for yourself. 76 00:03:48,419 --> 00:03:49,809 This is 16 times 16. 77 00:03:49,810 --> 00:03:53,159 So the square root of 256 is 16. 78 00:03:53,159 --> 00:03:56,400 So we can rewrite this whole thing as being equal to 79 00:03:56,400 --> 00:04:04,500 negative 2 plus 16 over negative 14. 80 00:04:04,500 --> 00:04:07,789 Or negative 2 minus-- right? 81 00:04:07,789 --> 00:04:10,179 This is plus 16 over negative 14. 82 00:04:10,180 --> 00:04:16,490 Or minus 16 over negative 14. 83 00:04:16,490 --> 00:04:20,740 If you think of it as plus or minus, that plus is that plus 84 00:04:20,740 --> 00:04:21,720 right there. 85 00:04:21,720 --> 00:04:26,510 And if you have that minus, that minus is 86 00:04:26,509 --> 00:04:28,250 that minus right there. 87 00:04:28,250 --> 00:04:29,910 Now we just have to evaluate these two numbers. 88 00:04:29,910 --> 00:04:32,430 89 00:04:32,430 --> 00:04:39,009 Negative 2 plus 16 is 14 divided by negative 14 is 90 00:04:39,009 --> 00:04:40,170 negative 1. 91 00:04:40,170 --> 00:04:42,470 So q could be equal to negative 1. 92 00:04:42,470 --> 00:04:47,430 Or negative 2 minus 16 is negative 18 divided by 93 00:04:47,430 --> 00:04:51,319 negative 14 is equal to 18 over 14. 94 00:04:51,319 --> 00:04:54,500 The negatives cancel out, which is equal to 9 over 7. 95 00:04:54,500 --> 00:04:57,189 So q could be equal to negative 1, or it could be 96 00:04:57,189 --> 00:04:59,500 equal to 9 over 7. 97 00:04:59,500 --> 00:05:02,100 And you could try these out, substitute these q's back into 98 00:05:02,100 --> 00:05:04,460 this original equation, and verify for yourself that they 99 00:05:04,459 --> 00:05:05,349 satisfy it. 100 00:05:05,350 --> 00:05:07,900 We could even do it with the first one. 101 00:05:07,899 --> 00:05:09,750 So if you take q is equal to negative 1. 102 00:05:09,750 --> 00:05:12,795 Negative 7 times negative 1 squared-- negative 1 squared 103 00:05:12,795 --> 00:05:17,210 is just 1-- so this would be negative 7 times 1, right? 104 00:05:17,209 --> 00:05:18,919 That's negative 1 squared. 105 00:05:18,920 --> 00:05:22,259 Negative 1 times 2 is minus 2 plus 9. 106 00:05:22,259 --> 00:05:26,149 So it's negative 7 minus 2, which is negative 9, plus 9, 107 00:05:26,149 --> 00:05:27,959 does indeed equal 0. 108 00:05:27,959 --> 00:05:29,180 So this checks out. 109 00:05:29,180 --> 00:05:31,780 And I'll leave it up to you to verify that 9 over 110 00:05:31,779 --> 00:05:34,149 7 also works out.