WEBVTT Kind: captions Language: en 00:00:01.766 --> 00:00:07.526 >> Welcome to the Cypress College Math Review on solving quadratic equations. 00:00:09.756 --> 00:00:12.666 There are four methods for solving quadratic equations. 00:00:13.986 --> 00:00:19.006 Solve by taking square roots, which is also called extracting square roots. 00:00:19.986 --> 00:00:23.626 Solve by factoring using the zero product rule. 00:00:24.476 --> 00:00:26.746 Solve by completing the square. 00:00:27.716 --> 00:00:30.616 And solve by using the quadratic formula. 00:00:31.336 --> 00:00:35.496 We are going to review each of these different methods. 00:00:37.766 --> 00:00:41.166 Objective one, solving by taking square roots. 00:00:41.846 --> 00:00:43.166 Let's review this method. 00:00:45.016 --> 00:00:50.136 First of all, in this equation we notice that the only X is in a square. 00:00:51.846 --> 00:00:55.766 This equation will solve very easily using this method. 00:00:56.886 --> 00:00:59.146 You want to isolate the perfect square. 00:00:59.506 --> 00:01:01.516 The perfect square is the X squared. 00:01:02.486 --> 00:01:05.276 Notice that there are no other Xs in the equation. 00:01:06.396 --> 00:01:09.146 So first we add 3 to both sides. 00:01:09.986 --> 00:01:14.236 Then we divide by 2. 00:01:14.866 --> 00:01:20.596 Now we have the perfect square completely isolated. 00:01:21.076 --> 00:01:24.566 The next step is to take the square root of both sides. 00:01:26.046 --> 00:01:38.476 Now on the left-hand side you get the absolute value of X which will give you plus or minus X. 00:01:41.476 --> 00:01:47.676 And when you solve for X you get plus or minus the square root of 5. 00:01:47.976 --> 00:01:50.076 No one actually shows those steps. 00:01:50.696 --> 00:01:55.516 Most people go straight from this equation, X squared equals 5, 00:01:56.116 --> 00:02:01.326 to X equals plus or minus the square root of 5. 00:02:01.946 --> 00:02:10.296 For this problem the perfect square is the square of X plus 4. 00:02:11.276 --> 00:02:13.776 Notice that there are no other Xs in the equation. 00:02:14.376 --> 00:02:16.306 So our first step is to isolate that. 00:02:19.416 --> 00:02:21.426 So we add 5 to both sides. 00:02:24.766 --> 00:02:26.416 Then divide both sides by 2. 00:02:30.556 --> 00:02:32.626 Now we take the square root of both sides. 00:02:33.256 --> 00:02:37.416 So X plus 4 is equal to plus or minus the square root of 12. 00:02:40.046 --> 00:02:43.246 We need to take out any perfect squares that we can out of 00:02:43.246 --> 00:02:46.826 that radical, and 4 is a perfect square. 00:02:52.360 --> 00:03:00.520 So we have our answer of X equals negative 4 plus or minus 2 times the square root of 3. 00:03:02.960 --> 00:03:04.680 Pause the video and try these problems. 00:03:08.840 --> 00:03:10.000 Objective two. 00:03:10.386 --> 00:03:11.446 Solve by factoring. 00:03:12.866 --> 00:03:17.366 In this equation we not only have the quadratic term 5X squared, 00:03:17.366 --> 00:03:20.946 but we also have the linear term negative 14X. 00:03:21.356 --> 00:03:24.726 So X is not just in that perfect square X squared. 00:03:25.776 --> 00:03:27.256 So our first method won't work. 00:03:28.036 --> 00:03:32.256 To solve by factoring, you need to move all terms to one side. 00:03:33.236 --> 00:03:36.396 I always like to have a positive leading coefficient 00:03:36.446 --> 00:03:39.836 when I'm factoring so I make sure that happens. 00:03:39.836 --> 00:03:41.816 I get a positive 5X squared. 00:03:42.866 --> 00:03:45.416 And the right-hand side you have to have a zero. 00:03:45.466 --> 00:03:46.836 It has to equal zero. 00:03:47.766 --> 00:03:52.106 The next step is to factor your trinomial. 00:03:52.766 --> 00:04:00.476 To get 5X squared I would have 5X and X. There's different options for 8. 00:04:00.806 --> 00:04:04.406 I could have 1 times 8 or 2 times 4. 00:04:05.116 --> 00:04:08.746 It will work out that I need a 4 here and a 2 there. 00:04:10.436 --> 00:04:16.256 This is 4X and this is 10X. 00:04:16.796 --> 00:04:18.646 Those would add to get 14X. 00:04:18.646 --> 00:04:22.876 The signs need to be the same because this is a positive. 00:04:22.876 --> 00:04:29.786 And they both need to be plus to be able to make it so that I have positive 14X. 00:04:30.946 --> 00:04:31.926 Now I have it factored. 00:04:32.276 --> 00:04:37.326 By the zero product rule we know that if A times B is equal to zero, 00:04:38.246 --> 00:04:41.596 then either A is equal to zero or B is equal to zero. 00:04:42.216 --> 00:04:48.036 So I have the quantity 5X plus 4 times the quantity X plus 2 is equal to 0. 00:04:48.586 --> 00:04:57.566 That means that either the first factor is equal to 0 or the second factor is equal to 0. 00:04:58.796 --> 00:05:01.406 Now we solve each of these individual equations. 00:05:03.706 --> 00:05:10.186 So we have X equals negative 4/5 or X equals negative 2, 00:05:10.346 --> 00:05:12.566 giving us the two answers to this equation. 00:05:16.446 --> 00:05:19.916 Our next example we will solve by factoring again. 00:05:20.116 --> 00:05:24.146 We want a positive leading coefficient so we move everything to the left. 00:05:25.736 --> 00:05:28.346 Equals 0. Now we factor. 00:05:28.966 --> 00:05:39.036 So we have 5X and 3X and we need a 2 here to give us 6X. 00:05:39.976 --> 00:05:43.726 And a 4 here to give us 20X. 00:05:44.946 --> 00:05:48.306 The signs are going to be different because the 8 is negative. 00:05:48.306 --> 00:05:58.956 And I'm going to need a negative 20X and a positive 6X to get the negative 14X that I need. 00:06:00.176 --> 00:06:02.356 Now the left-hand side is factored. 00:06:03.406 --> 00:06:05.266 Now we're going to use the zero product rule. 00:06:06.256 --> 00:06:13.016 Since A times B is equal to 0, either the first factor is equal to 0 00:06:14.646 --> 00:06:17.866 or the second factor is equal to 0. 00:06:18.646 --> 00:06:20.896 Then we solve each of these equations. 00:06:31.140 --> 00:06:41.680 So we get X equals negative 2/5 or X equals 4/3 which are the two solutions to this equation. 00:06:42.500 --> 00:06:44.080 Pause the video and try these problems. 00:06:49.180 --> 00:06:50.140 Objective three. 00:06:50.346 --> 00:06:52.106 Solve by completing the square. 00:06:52.586 --> 00:06:58.596 To solve by completing the square, we first move the constant term to the other side. 00:07:00.046 --> 00:07:02.386 So we add 3 to both sides for this equation. 00:07:03.286 --> 00:07:08.506 And then we divide by the leading coefficient so that you have a 1X squared. 00:07:08.506 --> 00:07:12.766 In this equation we already have 1X squared so we don't need to do that. 00:07:14.316 --> 00:07:16.106 Next you complete the square. 00:07:16.846 --> 00:07:21.656 To complete the square you take half of the negative 10 which is negative 5, 00:07:22.106 --> 00:07:24.896 square it, and add it to both sides. 00:07:25.966 --> 00:07:29.296 So to complete the square we have to add 25 to both sides. 00:07:29.296 --> 00:07:31.546 What you do to one side you have to do to the other. 00:07:32.216 --> 00:07:34.796 You're always adding when you complete the square. 00:07:36.166 --> 00:07:39.306 Next you write the left-hand side as a perfect square. 00:07:39.716 --> 00:07:40.346 X squared. 00:07:40.346 --> 00:07:43.226 This would be an X. This is minus so this is minus. 00:07:43.476 --> 00:07:45.366 And half of 10 was 5. 00:07:46.066 --> 00:07:46.656 Check it out. 00:07:46.656 --> 00:07:52.826 If you square X minus 5 you do get X squared minus 10X plus 25. 00:07:54.356 --> 00:07:56.476 The right-hand side is 28. 00:07:57.576 --> 00:07:58.916 Now we use our first method. 00:07:59.156 --> 00:08:00.806 We take the square root of both sides. 00:08:01.656 --> 00:08:05.746 So X minus 5 is equal to plus or minus the square root of 28. 00:08:06.316 --> 00:08:12.716 Now we simplify that radical and that's 4 times 7. 00:08:14.016 --> 00:08:19.446 So the right-hand side becomes plus or minus 2 times the square root of 7. 00:08:20.106 --> 00:08:23.176 We solve for X by adding 5 to both sides. 00:08:23.176 --> 00:08:26.876 So we get 5 plus or minus 2 times the square root of 7. 00:08:27.686 --> 00:08:30.066 Sometimes it's okay to write your answers like that. 00:08:31.636 --> 00:08:34.116 Other times your instructor wants them written separately. 00:08:34.426 --> 00:08:42.246 So 5 plus 2 times the square root of 7 or 5 minus 2 times the square root of 7. 00:08:42.536 --> 00:08:44.926 Those are the solutions to this equation. 00:08:50.556 --> 00:08:53.056 Let's solve this equation by completing the square. 00:08:55.056 --> 00:08:58.306 So the first step is to move the constant term to the other side. 00:08:59.376 --> 00:09:01.246 So we add 8 to both sides. 00:09:02.536 --> 00:09:07.796 The next step is to divide by the leading coefficient so that we have a 1X squared. 00:09:08.156 --> 00:09:13.226 Be sure you divide every single term on both sides of the equation by that coefficient. 00:09:14.496 --> 00:09:19.276 So we have X squared plus 6X is equal to 2. 00:09:21.006 --> 00:09:22.366 Now we complete the square. 00:09:23.016 --> 00:09:25.846 Half of 6 is 3, squared is 9. 00:09:26.406 --> 00:09:28.146 So we add 9 to both sides. 00:09:29.326 --> 00:09:32.176 The left-hand side becomes a perfect square. 00:09:32.456 --> 00:09:37.426 X squared becomes X plus, plus, half of 6 was 3. 00:09:38.536 --> 00:09:40.226 The right-hand side is 11. 00:09:41.896 --> 00:09:43.876 Now we take the square root of both sides. 00:09:44.396 --> 00:09:48.716 So X plus 3 equals plus or minus the square root of 11, 00:09:49.236 --> 00:09:51.636 and the square root of 11 cannot be simplified. 00:09:52.600 --> 00:09:58.180 So our answers are negative 3 plus or minus the square root of 11. 00:10:02.600 --> 00:10:05.240 Pause the video and try these problems. 00:10:09.340 --> 00:10:10.220 Objective four. 00:10:10.736 --> 00:10:12.706 Solve by using the quadratic formula. 00:10:13.236 --> 00:10:16.506 So for this equation we're going to solve using that method. 00:10:17.166 --> 00:10:20.776 So we first move all terms to one side and get them organized. 00:10:22.886 --> 00:10:24.826 So we add 12X to both sides. 00:10:25.846 --> 00:10:27.436 We have to have it equal to 0. 00:10:27.756 --> 00:10:34.676 So we must have it in the form AX squared plus BX plus C equals 0. 00:10:34.996 --> 00:10:42.746 From that we can determine that A is 3, B is 12, and C is 7. 00:10:44.316 --> 00:10:45.906 Next we write down the formula. 00:10:46.736 --> 00:10:57.336 So X equals negative B plus or minus the square root of B squared minus 4AC all over 2A. 00:10:59.146 --> 00:11:04.006 Use parentheses when you're replacing variables with numbers. 00:11:08.066 --> 00:11:14.296 So I have parentheses everywhere I see a letter. 00:11:14.456 --> 00:11:15.636 Now I substitute in. 00:11:16.676 --> 00:11:23.686 So B is 12, A is 3, and C is 7. 00:11:24.946 --> 00:11:26.336 Now we simplify this. 00:11:27.546 --> 00:11:37.976 So we have X equals negative 12 plus or minus the square root of -- And 12 squared is 144. 00:11:38.806 --> 00:11:41.336 And then minus 84. 00:11:41.586 --> 00:11:45.016 Then that's all over 6. 00:11:45.936 --> 00:11:52.836 So we get negative 12 plus or minus the square root of 60 all over 6. 00:11:52.886 --> 00:11:55.906 Be careful not to cancel the 12 and the 6. 00:11:56.336 --> 00:12:00.366 You can only cancel factors, and the numerator is not factored. 00:12:01.786 --> 00:12:03.926 We have to simplify that radical. 00:12:05.026 --> 00:12:08.976 60 is 4 times 15. 00:12:11.036 --> 00:12:17.546 So we have 2 times the square root of 15. 00:12:19.986 --> 00:12:21.666 Now we can factor out a 2. 00:12:22.276 --> 00:12:31.726 We do so and we end up with negative 6 plus or minus the square root of 15 all over 6. 00:12:32.646 --> 00:12:33.756 Now we can cancel. 00:12:34.446 --> 00:12:37.886 2 goes in to here once and 2 goes in to here 3 times. 00:12:40.176 --> 00:12:53.526 So our answers to this equation are negative 6 plus or minus the square root of 15 all over 3. 00:13:01.716 --> 00:13:04.506 We're going to solve this using the quadratic formula again. 00:13:05.616 --> 00:13:08.066 So we move all terms to one side. 00:13:09.846 --> 00:13:11.036 Have it equal to 0. 00:13:13.016 --> 00:13:19.026 So we have it in the form AX squared plus BX plus C equals 0. 00:13:19.906 --> 00:13:27.356 That tells us that A is 5, B is negative 8, and C is negative 3. 00:13:28.336 --> 00:13:29.726 Next we write the formula. 00:13:30.486 --> 00:13:41.176 X equals negative B plus or minus the square root of B squared minus 4AC all over 2A. 00:13:42.406 --> 00:13:47.806 Then we use parentheses since we're replacing variables with numbers. 00:13:50.406 --> 00:14:05.566 So we have that B is negative 8, A is 5, and C is negative 3. 00:14:07.206 --> 00:14:08.756 Now we simplify this. 00:14:10.436 --> 00:14:18.566 So we have positive 8 plus or minus the square root of 64 00:14:18.566 --> 00:14:23.346 because that's negative 8 quantity squared, and then watch out for your signs. 00:14:23.346 --> 00:14:28.366 We've got two negatives so we end up with plus 60 all over 10. 00:14:29.566 --> 00:14:37.676 So we have X equals 8 plus or minus the square root of 124 all over 10. 00:14:38.126 --> 00:14:46.376 124 is 2 times 62 and 62 is 2 times 31. 00:14:48.496 --> 00:14:52.976 So X equals 8 plus or minus the square root of 4 00:14:52.976 --> 00:14:58.106 because we have 2 squared times the 31 all over the 10. 00:14:59.596 --> 00:15:01.146 I'm trying to simplify the radical. 00:15:01.536 --> 00:15:06.946 So we get 8 plus or minus 2 rad 31 all over 10. 00:15:07.866 --> 00:15:15.556 Now we factor out a 2 and we're left with 4 plus or minus the square root of 31 all over 10. 00:15:16.006 --> 00:15:17.506 Now you can cancel. 00:15:18.496 --> 00:15:23.506 So many people coming back over here will cancel with the 8 and the 10. 00:15:23.506 --> 00:15:29.456 And you cannot cancel the 8 with the 10 because the 8 is not a factor of the numerator. 00:15:30.076 --> 00:15:33.906 Looking over here on the right, the 2 is a factor of the numerator. 00:15:34.396 --> 00:15:41.076 Since 2 is a factor of the numerator, I can cancel and I get 1 over 5. 00:15:42.840 --> 00:15:51.700 So the solution to this equation is 4 plus or minus the square root of 31 all over 5. 00:15:57.720 --> 00:16:00.360 Pause the video and try these problems. 00:16:03.740 --> 00:16:04.920 Objective five. 00:16:05.586 --> 00:16:08.616 Determine the best method to solve a quadratic equation. 00:16:10.446 --> 00:16:18.156 In the equation AX squared plus BX plus C equals 0, AX squared is a quadratic term, 00:16:19.176 --> 00:16:23.496 BX is called the linear term, and C is the constant. 00:16:27.396 --> 00:16:32.276 If your equation does not have a constant term, then solve by factoring. 00:16:33.346 --> 00:16:35.706 That's definitely going to be the easiest method. 00:16:36.666 --> 00:16:42.146 In this equation we move all terms to one side and we notice that there's no constant term. 00:16:42.556 --> 00:16:45.436 Every single term has an X in it. 00:16:45.616 --> 00:16:51.576 So we factor, set each factor equal 0, and then solve. 00:16:52.136 --> 00:16:58.116 Any time the equation factors easily then that's the easiest method to solve. 00:16:58.116 --> 00:17:01.846 Solve by factoring if the equation factors easily. 00:17:06.456 --> 00:17:12.326 If your equation does not have a linear term, then solve by taking square roots. 00:17:13.016 --> 00:17:17.456 The only place you see the X is in the perfect square X squared. 00:17:18.776 --> 00:17:21.246 So you're going to solve by taking square roots. 00:17:22.526 --> 00:17:27.766 So in this equation we move 9 to the other side, divide by 4, 00:17:28.416 --> 00:17:30.346 and take the square root of both sides. 00:17:30.606 --> 00:17:35.146 So our answers are X equals plus or minus 3/2. 00:17:38.096 --> 00:17:46.996 If your equation looks like 1X squared plus BX plus C equals 0, and B is even, 00:17:47.486 --> 00:17:50.226 then completing the square is really fast. 00:17:50.876 --> 00:17:58.816 Any time you have a 1X squared and you have an even linear coefficient, B is even, 00:18:00.756 --> 00:18:02.526 then you should complete the square. 00:18:03.446 --> 00:18:09.836 So for this equation we have 1X squared minus 8X, and then we have a constant term. 00:18:09.916 --> 00:18:14.926 The constant term can be on either side, but definitely for this equation solve 00:18:14.926 --> 00:18:17.406 by completing the square is the easiest. 00:18:18.686 --> 00:18:20.906 So we have the constant on the other side. 00:18:21.626 --> 00:18:24.226 That's the first step, already done for us. 00:18:25.116 --> 00:18:26.706 Now we complete the square. 00:18:27.136 --> 00:18:30.386 Half of 8 is 4, squared is 16. 00:18:30.666 --> 00:18:32.486 We add 16 to both sides. 00:18:33.086 --> 00:18:36.346 The left-hand side we write as a perfect square. 00:18:38.166 --> 00:18:41.116 Then we take the square root of both sides of the equation. 00:18:41.236 --> 00:18:45.906 So we have X minus 4 equals plus or minus the square root of 19. 00:18:46.856 --> 00:18:52.956 We add 4 to both sides and we get our solutions, 4 plus or minus the square root of 19. 00:18:57.366 --> 00:19:03.716 If your equation looks like this where the only place you see the X is in a perfect square, 00:19:04.876 --> 00:19:08.416 then solving by taking square roots is the easiest method. 00:19:10.076 --> 00:19:13.566 So we want to get the perfect square by itself by adding 4 00:19:13.566 --> 00:19:16.666 to both sides and then dividing by 3. 00:19:17.466 --> 00:19:19.956 Then we take the square root of both sides. 00:19:20.746 --> 00:19:27.976 So we get X plus 2 equals plus or minus the square root of 4, but the square root of 4 is 2 00:19:28.956 --> 00:19:33.216 so now we have to write it as two separate equations because our radical disappeared. 00:19:34.256 --> 00:19:40.366 We subtract 2 from both sides and we get X equals negative 2 plus 2 which is 0 00:19:41.026 --> 00:19:44.956 or X equals negative 2 minus 2 which is negative 4, 00:19:45.386 --> 00:19:48.336 and those are the two solutions to this equation. 00:19:51.236 --> 00:19:58.466 If your equation involves fractions, then multiply both sides of the equation 00:19:58.466 --> 00:20:02.056 by the appropriate quantity, which of course is the least common denominator, 00:20:03.116 --> 00:20:04.766 to get rid of the fractions. 00:20:05.666 --> 00:20:09.836 Then use whatever method is easiest to solve the resulting equation. 00:20:11.056 --> 00:20:13.416 If your equation involves decimals, 00:20:14.106 --> 00:20:17.166 then the quadratic formula is probably the easiest method. 00:20:21.796 --> 00:20:26.046 If you're working on a word problem or any problem where you're allowed 00:20:26.086 --> 00:20:31.606 to use an approximation, then the quadratic formula is usually the fastest. 00:20:32.236 --> 00:20:35.096 Substitute A, B, and C in to the formula, 00:20:35.496 --> 00:20:38.656 then pull out a calculator and approximate the answers. 00:20:39.100 --> 00:20:41.936 No need to simplify. 00:20:43.420 --> 00:20:45.280 Pause the video and try these problems. 00:20:49.140 --> 00:20:50.620 Objective six. 00:20:51.146 --> 00:20:52.836 Practice solving by any method. 00:20:54.196 --> 00:20:58.366 For each of the problems in the exercise set, choose the easiest method, 00:20:59.540 --> 00:21:04.080 then solve the quadratic equations. 00:21:05.320 --> 00:21:07.860 Pause the video and try these problems.