WEBVTT Kind: captions Language: en 00:00:01.166 --> 00:00:07.076 >> Welcome to the Cypress College Math Review on Completing the Square. 00:00:07.236 --> 00:00:10.126 Our first objective deals with the square root property. 00:00:10.576 --> 00:00:13.766 Before we get to what the square root property is we want 00:00:13.766 --> 00:00:17.756 to review first what a quadratic equation in standard form looks like. 00:00:18.226 --> 00:00:23.886 Suppose a, b, and c are real numbers such that a does not equal to zero. 00:00:24.676 --> 00:00:26.456 Then any equation that could be written 00:00:26.456 --> 00:00:33.256 in the form ax squared plus bx plus c equals zero is considered 00:00:33.256 --> 00:00:36.126 to be a quadratic equation in standard form. 00:00:36.556 --> 00:00:42.926 The quadratic part is because you have a term that has its variable with an exponent of two. 00:00:43.496 --> 00:00:46.126 What makes it standard form is that it's equaling zero. 00:00:47.006 --> 00:00:50.736 So with that in mind let's now look at what is the square root property. 00:00:51.476 --> 00:00:53.726 Suppose k is a real number. 00:00:54.426 --> 00:01:02.546 If x squared equals k, then x equals the square root of k or x equals negative square root 00:01:02.546 --> 00:01:09.646 of k. This may also be written as x equals plus or minus the square root of k. 00:01:10.256 --> 00:01:15.166 So we will use the plus or minus symbol to represent the positive 00:01:15.166 --> 00:01:17.486 and the negative version of our square root. 00:01:17.836 --> 00:01:21.916 While we still need to simplify the radical we now have some notation that's a little bit 00:01:21.916 --> 00:01:22.706 more compact. 00:01:24.476 --> 00:01:28.726 Now let us solve some quadratic equations actually using the square root property. 00:01:29.026 --> 00:01:34.116 In our first example we have x squared plus 3 equals negative 15. 00:01:34.736 --> 00:01:37.876 Before you can actually use the square root property you need 00:01:37.876 --> 00:01:40.036 to isolate the square term first. 00:01:40.176 --> 00:01:45.606 So here x squared is that square term and in order to get it alone on the left hand side 00:01:45.606 --> 00:01:49.076 of the equation we need to subtract three form both sides. 00:01:49.506 --> 00:01:53.546 This leaves us with x squared equaling negative 18. 00:01:54.266 --> 00:01:58.996 Once we have x squared alone we are now ready to use the square root property, 00:01:58.996 --> 00:02:06.006 which says x sill be equal to plus or minus the square root of negative 18. 00:02:07.256 --> 00:02:11.206 Once we have done all this we can now focus on simplifying the radical. 00:02:11.516 --> 00:02:15.256 One of the first things we can do when it comes to simplifying the square root 00:02:15.256 --> 00:02:20.906 of negative 18 is remember that i equals the square root of negative 1. 00:02:21.556 --> 00:02:26.626 So we can actually have that negative inside of our radical leave as i. 00:02:26.996 --> 00:02:33.546 So we can now rewrite our statement as x equals plus or minus i times the square root of 18. 00:02:34.456 --> 00:02:38.646 Now we can focus all of our attention on just simplifying the square root of 18. 00:02:38.846 --> 00:02:44.236 Our goal is to try to find what perfect squares make up the number 18. 00:02:44.636 --> 00:02:48.906 So let us simplify the radical by noticing x equals plus 00:02:48.906 --> 00:02:53.106 or minus i times the square root of 2 times 9. 00:02:53.456 --> 00:02:57.996 The reason why we may want to show our statement this way is so that we can see 00:02:58.216 --> 00:03:03.746 that we have the perfect square nine inside of our square root and we know 00:03:03.746 --> 00:03:09.616 since the square root of 9 is 3 we can simplify by saying x equals plus 00:03:09.616 --> 00:03:12.786 or minus 3 i times the square root of 2. 00:03:13.566 --> 00:03:17.216 Since we can't simplify any further we're now ready to write our solution set 00:03:17.606 --> 00:03:24.126 which can be written as an open brace, and from here we just list our solutions that we found. 00:03:24.506 --> 00:03:30.596 So we write negative 3 i times the square root of 2 and positive 3 i times the square root 00:03:30.596 --> 00:03:33.166 of 2, and then we close it with a brace. 00:03:36.916 --> 00:03:38.076 Let's try another example. 00:03:38.706 --> 00:03:43.446 In this one we're going to notice that our square term is no longer just a single letter, 00:03:43.446 --> 00:03:47.536 it actually contains an expression so let's read what we have to look at. 00:03:47.716 --> 00:03:55.416 We have here the quantity 4 m plus 5 being squared, then we subtract 8 equaling 20. 00:03:56.006 --> 00:03:59.876 So our goal is still the same as it was before; we can't use the square root property 00:04:00.046 --> 00:04:02.126 until we isolate the squared term. 00:04:02.566 --> 00:04:08.926 Here 4 m plus 5 is the quantity that is being squared so that is what we want to get alone. 00:04:09.246 --> 00:04:11.686 First step we need to add 8 to both sides. 00:04:11.986 --> 00:04:17.616 This leaves us with 4 m plus 5 quantity squared equals 28. 00:04:18.466 --> 00:04:23.046 Here we can finally use the square root property which has the inside 00:04:23.046 --> 00:04:25.066 of what was being squared staying the same, 00:04:25.186 --> 00:04:30.336 4 m plus 5 equals plus or minus the square root of 28. 00:04:31.486 --> 00:04:35.186 Before we focus our attention on solving for m on the left hand side 00:04:35.186 --> 00:04:38.786 of the equation let's simplify the radical first, this will just allow it 00:04:38.786 --> 00:04:41.496 so that we don't forget to do it later on and we're going to have 00:04:41.496 --> 00:04:43.406 to do it anyway so let's just do it right now. 00:04:44.046 --> 00:04:51.786 Let's keep 4 m plus 5 on the left equaling plus or minus the square root of 4 times 7. 00:04:52.246 --> 00:04:56.806 Why we're writing 28 in this way is because we notice 4 is a perfect square 00:04:56.866 --> 00:05:02.876 because the square root of 4 is 2, leaving the 7 left over inside of our square root. 00:05:03.056 --> 00:05:07.676 So now we can rewrite our statement as 4 m plus 5 equals plus 00:05:07.676 --> 00:05:10.496 or minus 2 times the square root of 7. 00:05:11.236 --> 00:05:15.346 Since we can't simplify the radical any further we can now focus the rest 00:05:15.346 --> 00:05:17.506 of our attention on solving for the variable. 00:05:18.216 --> 00:05:21.866 Here the first thing we need to do will be to subtract 5 from both sides 00:05:21.866 --> 00:05:28.376 so we get 4 m equaling negative 5 plus or minus 2 times the square root of 7. 00:05:29.346 --> 00:05:33.866 After that we want to divide both sides of our equation by the number 4, 00:05:34.106 --> 00:05:38.536 which leaves us with m equals negative 5 plus 00:05:38.536 --> 00:05:43.196 or minus 2 times the square root of 7 all being divided by 4. 00:05:44.846 --> 00:05:49.786 Once we have this we are now ready to write our final answer as a solution set. 00:05:50.186 --> 00:05:59.046 So our solution set will take the form of open brace negative 5 minus 2 times the square root 00:05:59.046 --> 00:06:03.376 of 7 all divided by 4, and we have our second solution 00:06:03.376 --> 00:06:09.920 of negative 5 plus 2 times the square root of 7 all divided by 4; then we close it with a brace. 00:06:16.560 --> 00:06:21.360 Pause the video and try these problems. 00:06:25.440 --> 00:06:28.960 In objective two we're going to review perfect square trinomials. 00:06:29.516 --> 00:06:33.936 A perfect square trinomial is a trinomial which is a three term polynomial 00:06:34.426 --> 00:06:38.496 that when you factor it results in a binomial being squared. 00:06:38.746 --> 00:06:46.176 So if you have a trinomial of the form a squared plus 2 ab plus b square it will factor 00:06:46.176 --> 00:06:49.856 as a binomial a plus b quantity squared. 00:06:51.066 --> 00:06:56.146 If instead of addition for your middle term you actually have subtraction the factorization 00:06:56.146 --> 00:06:59.336 changes to a minus b quantity squared. 00:07:00.216 --> 00:07:03.546 The way that you can remember this is that the sign in the middle 00:07:03.546 --> 00:07:07.996 of the trinomial always match the sign in the middle of your binomial when it's factored. 00:07:08.556 --> 00:07:12.856 Let us try some examples. 00:07:13.236 --> 00:07:14.936 Factor the perfect square trinomial. 00:07:14.936 --> 00:07:19.116 In part a we have x squared plus 18 x plus 81. 00:07:19.316 --> 00:07:24.466 Well we first want to verify does this even have the structure of a perfect square trinomial? 00:07:24.866 --> 00:07:30.396 How we can go about doing this is setting up some parenthesis that actually shows the formula 00:07:30.396 --> 00:07:32.816 for a perfect square trinomial and if it checks 00:07:32.816 --> 00:07:36.266 out we can then factor it using those pieces that we found. 00:07:36.726 --> 00:07:39.296 So let's begin by writing some parenthesis, 00:07:39.296 --> 00:07:42.226 leave some space inside, and put a square on the outside. 00:07:42.936 --> 00:07:47.616 Then we'll write a plus 2 times and we'll write two more sets of parenthesis, 00:07:47.616 --> 00:07:53.306 leaving the insides blank for now, plus a set of parenthesis squared. 00:07:53.946 --> 00:07:57.206 Now we're going to go in and fill all these blanks. 00:07:57.726 --> 00:08:01.846 The first set of parenthesis we are going to ask ourselves what would I need to square in order 00:08:01.846 --> 00:08:05.746 to get the first term of x squared in the original trinomial. 00:08:06.386 --> 00:08:10.536 That would have to be a x because x when you square it gives you x squared. 00:08:11.436 --> 00:08:14.516 Now before we check the middle I always encourage people to go 00:08:14.516 --> 00:08:16.576 to the back and let's check the last term. 00:08:17.166 --> 00:08:21.096 Let's see there that what would you have to square in order to get 81. 00:08:21.756 --> 00:08:26.006 Well 9. So now that you see that the first term and the last term are matching 00:08:26.006 --> 00:08:31.436 up let's take what was in those two sets of parenthesis and put them into the middle set 00:08:31.436 --> 00:08:38.336 of parenthesis and we're going to check is it true that 2 times x times 9 does it equal 18 x, 00:08:38.646 --> 00:08:40.116 and if the answer is yes like it is 00:08:40.116 --> 00:08:44.006 in this example then what we have is actually a perfect square trinomial, 00:08:44.346 --> 00:08:51.576 which means we can now factor this by taking our x, our 9, and since the sign in the middle 00:08:51.576 --> 00:08:55.486 of the original trinomial was a plus sign we also will have a plus sign. 00:08:55.486 --> 00:09:00.976 So now we can write our factorization as x plus 9 quantity squared. 00:09:06.656 --> 00:09:10.736 For our next example we're not just going to have the leading coefficient being 1, 00:09:10.736 --> 00:09:13.036 what happens if the leading coefficient is another number? 00:09:13.376 --> 00:09:20.266 So in example b we have 9 c squared minus 66 c plus 121. 00:09:20.876 --> 00:09:24.406 We're going to set up everything the same way as we did before but now it's just going 00:09:24.406 --> 00:09:28.676 to be a matter of what are the pieces that we need to create that above trinomial. 00:09:29.156 --> 00:09:34.226 So let's set up our actual parenthesis, so we equal a set of parenthesis squared. 00:09:35.116 --> 00:09:38.486 Since we have subtraction of that 66 c we're going 00:09:38.486 --> 00:09:43.256 to write minus 2 times we'll write our two sets of parenthesis, 00:09:43.956 --> 00:09:46.986 plus a set of parenthesis squared. 00:09:48.006 --> 00:09:49.616 Now we're going to check that first term. 00:09:49.616 --> 00:09:54.566 In order to get 9 c squared wouldn't you have to use a 3 c? 00:09:55.536 --> 00:10:01.376 To get that 121 for the last term wouldn't you have to square 11? 00:10:02.086 --> 00:10:08.616 So now we can take 3 c, now we can take our 11, fill in the middle two sets of parenthesis 00:10:08.876 --> 00:10:14.316 and check is it true that negative 2 times 3 c times 11 does it 00:10:14.316 --> 00:10:22.216 in fact equal negative 66 c. Well negative 2 times 3 c would give you negative 6 c 00:10:22.436 --> 00:10:28.086 and negative 6 c when you multiply by 11 does in fact give you negative 66 c. 00:10:28.556 --> 00:10:33.606 So again we have a perfect square trinomial so we can take the 3 c and the 11, 00:10:33.786 --> 00:10:40.796 we can write them now taking the subtraction with us to get 3 c minus 11 quantity squared. 00:10:45.376 --> 00:10:48.806 We are now ready for completing a perfect square trinomial. 00:10:49.286 --> 00:10:54.086 We can make x squared plus bx a perfect square trinomial by adding 00:10:54.086 --> 00:10:58.106 to it 1/2 times b and then we square that result. 00:10:58.956 --> 00:11:03.326 The b in this formula is coming from the number in front of x; 00:11:03.326 --> 00:11:05.436 this is known as the linear coefficient. 00:11:06.576 --> 00:11:11.396 This is only possible though when the leading coefficient, the number that's in front 00:11:11.396 --> 00:11:16.416 of the x squared, is positive 1, so we will have to be aware of that when we start playing 00:11:16.416 --> 00:11:17.766 with different examples in the future. 00:11:18.846 --> 00:11:22.046 Let's start now with an example of this. 00:11:23.166 --> 00:11:27.886 Find what number must be added to the expression in order to make it a perfect square trinomial. 00:11:28.156 --> 00:11:29.656 Then factor the trinomial. 00:11:30.526 --> 00:11:40.376 We have x squared minus 8 x. The b value we have is negative 8, so we will take 1/2 multiply it 00:11:40.406 --> 00:11:43.546 by negative 8 and we will square that result. 00:11:44.806 --> 00:11:49.226 We could jump straight to the simplified answer here but it's actually going 00:11:49.226 --> 00:11:51.216 to benefit us by showing the middle step. 00:11:51.436 --> 00:11:55.256 This will allow us to use a shortcut when it comes to that factoring step. 00:11:55.676 --> 00:11:59.496 So let's take half of negative 8, which is negative 4, 00:12:00.046 --> 00:12:04.596 we'll square it; so negative 4 squared equals 16. 00:12:05.176 --> 00:12:10.966 Sixteen is the number that is going to complete the perfect square trinomial for our expression, 00:12:11.156 --> 00:12:15.336 so we write plus 16, and now we're ready to factor. 00:12:15.646 --> 00:12:18.026 But when it comes to factoring this is where we can take advantage 00:12:18.026 --> 00:12:19.836 of that shortcut that I've mentioned earlier. 00:12:20.226 --> 00:12:25.476 Here we can write the letter of the problem, which is x, and the number you need 00:12:25.566 --> 00:12:30.086 to factor this will be the number we had before we squared it. 00:12:30.376 --> 00:12:34.416 So since we had negative 4 we're going to write minus 4. 00:12:35.246 --> 00:12:39.116 We will put this in parenthesis and then put a power of 2 on the outside. 00:12:39.116 --> 00:12:42.666 So we have x minus 4 quantity squared. 00:12:43.286 --> 00:12:45.886 If we really wanted to check this multiplying 00:12:45.886 --> 00:12:51.960 out x minus 4 quantity squared will get us the x squared minus 8 x plus 16. 00:12:56.760 --> 00:12:59.920 Pause the video and try these problems. 00:13:09.040 --> 00:13:12.480 Objective three, completing the square when a equals 1. 00:13:13.166 --> 00:13:17.066 In this example we're going to solve the quadratic equation by completing the square. 00:13:17.306 --> 00:13:22.456 What we have is x squared minus 4 x minus 10 equals zero. 00:13:23.296 --> 00:13:26.796 Since the leading coefficient is already 1 all we need 00:13:26.796 --> 00:13:29.776 to do first is isolate the variable terms. 00:13:30.126 --> 00:13:35.126 This means get x squared minus 4 x alone on one side of the equal sign. 00:13:35.576 --> 00:13:39.236 Well in order to do this we need to add 10 to both sides. 00:13:39.746 --> 00:13:44.736 So by doing so we end up getting x squared minus 4 x equaling 10. 00:13:45.576 --> 00:13:49.246 Once we have the variable terms alone on one side we're ready 00:13:49.246 --> 00:13:51.106 to do that completing the square step. 00:13:51.106 --> 00:13:56.086 We want to try to figure out what number we need to add to our expression on the left in order 00:13:56.086 --> 00:13:58.146 to make it a perfect square trinomial. 00:13:58.506 --> 00:14:03.146 This is where we use 1/2 times b and we square that result. 00:14:03.636 --> 00:14:09.036 So to complete the perfect square trinomial in this problem we take 1/2 and we multiply it 00:14:09.096 --> 00:14:14.246 by the b value here, which is negative 4, and we will square our result. 00:14:15.186 --> 00:14:18.456 In order to still use the shortcut we will show the simplifying step, 00:14:18.626 --> 00:14:24.996 1/2 times negative 4 is just negative 2, which we will square to get positive 4. 00:14:26.096 --> 00:14:29.986 Four is the number that we want to now add to both sides of our equation. 00:14:30.646 --> 00:14:37.126 This leads us to x squared minus 4 x plus 4 equaling 10 plus 4. 00:14:38.026 --> 00:14:43.246 From here we're ready to factor the perfect square trinomial that we created on the left 00:14:43.246 --> 00:14:46.166 and all we need to do to the right is simplify the numbers. 00:14:46.786 --> 00:14:50.626 To factor the perfect square trinomial we will write x, 00:14:51.176 --> 00:14:57.246 and since it was negative 2 before we squared it that's what we'll write next, minus 2, 00:14:57.996 --> 00:15:03.016 put parenthesis around this so that we can square the quantity, and what we will be equal 00:15:03.016 --> 00:15:06.646 to is 10 plus 4, which is just 14. 00:15:11.406 --> 00:15:15.306 Now that everything has been factored and simplified we are now ready 00:15:15.306 --> 00:15:18.446 to use the square root property to solve for the variable. 00:15:19.286 --> 00:15:27.146 To do this we write x minus 2 equals plus or minus the square root of 14. 00:15:28.336 --> 00:15:34.116 Since 14 does not contain any perfect square factors we leave our radical 00:15:34.116 --> 00:15:35.666 as the square root of 14. 00:15:36.266 --> 00:15:39.436 All we need to do now is add 2 to both sides of our equation 00:15:39.436 --> 00:15:45.276 so that we get x equals 2 plus or minus the square root of 14. 00:15:46.136 --> 00:15:49.776 And our final step is to use solution sets to represent our answers. 00:15:50.496 --> 00:15:57.256 So we write an open brace, 2 minus the square root of 14, we'll write our comma, 00:15:58.160 --> 00:16:02.800 and then we write 2 plus the square root of 14, and close it with a brace. 00:16:07.280 --> 00:16:10.520 Pause the video and try these problems. 00:16:19.040 --> 00:16:23.240 Objective four, completing the square when a does not equal 1. 00:16:23.716 --> 00:16:27.516 In this example we're going to solve the quadratic equation by completing the square 00:16:27.996 --> 00:16:32.986 but you're going to notice that our leading coefficient is no longer a positive 1. 00:16:33.126 --> 00:16:39.106 Here we have 2 x squared plus 3 x minus 52 equaling zero. 00:16:39.106 --> 00:16:44.496 We're going to start of the same way, let's isolate the variable terms first. 00:16:44.906 --> 00:16:48.776 To do this we'll just add 52 to both sides of our equation. 00:16:49.356 --> 00:16:54.336 This leads us to 2 x squared plus 3 x equaling 52. 00:16:55.436 --> 00:17:02.176 From here we want to turn 2 x squared into just 1 x squared; in order to do this we're going 00:17:02.176 --> 00:17:09.006 to divide every term by the leading coefficient a in the case when a is not equaling 1. 00:17:09.516 --> 00:17:14.716 So in our problem we're going to divide every term on both sides of the equation by 2. 00:17:15.386 --> 00:17:21.306 So we'll divide 2 x squared by 2, we'll divide 3 x by 2, and we'll divide 52 by 2. 00:17:22.146 --> 00:17:31.206 When we reduce each of the fractions as best we can we get x squared plus 3/2 x equaling 26. 00:17:32.336 --> 00:17:36.416 Now we want to complete the perfect square trinomial and solve 00:17:36.416 --> 00:17:38.356 for the variable as we did before. 00:17:38.896 --> 00:17:43.636 This is where we're going to use 1/2 times b and then square the result. 00:17:44.486 --> 00:17:50.486 So in our problem we're going to take 1/2, we multiply it by the b value 3/2, 00:17:50.636 --> 00:17:52.226 and then we'll square that result. 00:17:52.936 --> 00:17:59.986 Simplifying gets us 3/4 squared, and then once we square we get 9/16. 00:18:04.586 --> 00:18:08.676 Nine sixteenths is now the number we're going to add to both sides of our equation, 00:18:09.086 --> 00:18:20.666 so we write x squared plus 3/2 x plus 9/16 equals 26 plus 9/16. 00:18:21.836 --> 00:18:26.686 Once we've done this we can now factor the perfect square trinomial that is on the left. 00:18:26.946 --> 00:18:30.376 This is absolutely when we want to take advantage of our shortcut. 00:18:30.806 --> 00:18:40.376 Here we will write x, and since we had positive 3/4 we write plus 3/4 quantity squared. 00:18:41.366 --> 00:18:46.046 On the right we are going to have to do some fraction addition where we will need 00:18:46.046 --> 00:18:52.666 to find a common denominator and then rewrite the 26 plus 9/16 as a single fraction. 00:18:53.456 --> 00:18:57.966 After we have done this work we end up getting 425/16. 00:18:58.846 --> 00:19:01.666 We are now ready to use the square root property. 00:19:02.386 --> 00:19:08.196 To do it we're going to get x plus 3/4 equals plus 00:19:08.196 --> 00:19:13.546 or minus the square root of 425 divided by 16. 00:19:15.036 --> 00:19:20.276 Before we move the 3/4 over, as always, let us simplify the radical first 00:19:20.276 --> 00:19:21.706 so that we don't forget to do it. 00:19:22.536 --> 00:19:26.766 We have a property of radicals that says whenever you have a quotient inside 00:19:26.766 --> 00:19:30.066 of a radical you can split it apart so that you're taking the root 00:19:30.066 --> 00:19:32.716 of both the numerator and denominator separately. 00:19:33.566 --> 00:19:35.836 The reason why we do this is because we want 00:19:35.836 --> 00:19:39.186 to make sure we don't have any radicals left over in a denominator. 00:19:39.766 --> 00:19:47.826 In our problem we're going to have x plus 3/4 equals plus or minus the square root 00:19:48.226 --> 00:19:53.416 of 425 divided by the square root of 16. 00:19:53.956 --> 00:19:59.566 We can now simplify the numerator to get 5 times the square root of 17 00:20:00.006 --> 00:20:02.686 and we can simplify the denominator to get 4. 00:20:03.306 --> 00:20:08.886 This leaves us with x plus 3/4 equals plus 00:20:08.886 --> 00:20:14.626 or minus 5 times the square root of 17 divided by 4. 00:20:15.156 --> 00:20:20.736 We will now solve for x by subtracting 3/4 from both sides of our equation. 00:20:21.256 --> 00:20:26.646 This leads to x equals negative 3/4 plus 00:20:26.646 --> 00:20:32.366 or minus 5 times the square root of 17 divided by 4. 00:20:33.416 --> 00:20:37.886 With x being solved we can now write our final solution set. 00:20:38.186 --> 00:20:45.986 So we have negative 3/4 minus 5 times the square root of 17 divided by 4 00:20:46.640 --> 00:20:53.280 and we have negative 3/4 plus 5 times the square root of 17 divided by 4. 00:21:00.880 --> 00:21:03.520 Pause the video and try these problems.