Algebra

Solving quadratic equations by completing the square | Algebra II | Khan Academy

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In this presentation, I will show you a method called Complete the box What is perfect in this technique is that it will work for any Quadratic equation, which is actually a foundation Quadratic formula And in the next show or the following presentation, I will prove Quadratic formula using square completions But before doing that, we will need to understand Everything about him And it actually consists of what we did last Display, where we solve the quadratic equations Using complete the box Let’s assume I have a square equation x ^ 2 – 4x = 5 I will put this void for a specific reason In the last show, we saw that this could To solve directly if the left side Whole square As you can see, completing the box is a conversion Quadratic equation to square whole, to engineer it Add and subtract from both sides to become Whole square How can we do that? Well, so that the left side becomes an entire square There must be a number We should have a number here so that if you do Squared, I would get that number, if I had 2 x Number, get -4 Remember that, I thought it would be It is clear through several examples I want x ^ 2 – 4x + something to equal x – a ^ 2 We don’t know what a’s worth yet, but we are We know many things When I square things – this will be x ^ 2 – 2a + a ^ 2 So if you look at this pattern here, that should be Sorry, x ^ 2 – 2ax– this should be 2ax This would be a ^ 2 So this number, that is, a will be half -4, that is, a must To be -2, right? Because 2 x a is -4 a = -2, and if a = -2, what is the value of a ^ 2? Well, a ^ 2 is going to be positive 4 Perhaps all of this seems complicated now But I will show you the rationale You are looking at this parameter here And you say, well, what is half of that parameter? Well, half of that coefficient is -2 We can say that a = -2 – the same idea here– And then you square it That is, you squared a, and you get positive 4 So we add positive 4 here We add 4 Now, according to what we did in the first equation You must know that you cannot do anything for only one side From the equation You cannot add 4 to one side of the equation If x ^ 2 – 4x equals 5, so when we add 4 Will not equal 5 It will equal 5 + 4 We add 4 to the left side because we want it It becomes a whole square But if we add something to the left side, we should To add it to the right side And now, we’ve got an issue Similar to the issues that we solved in the last presentation What is this left side? Let me rewrite everything We now have x ^ 2 – 4x + 4 = 0 All we did was add 4 to both sides of the equation But we added 4 to a specific goal, which is to become the left side Whole square What is this now? What number when I multiply it by himself will be the sum of 4 and When I combine it with himself it will be -2? Well, we have already answered this question He -2 So we get (x – 2) (x – 2) = 0 Or we can skip this step and write (x – 2) ^ 2 = 0 And then we take the square root of both sides, and we get x – 2 = + or – 3 We add 2 to both sides, and we get x equal to 2 + or – 3 This tells us that x can be equal to 2 + 3, i.e. 5 Or x equals 2-3, i.e. And we’re done I want to be clear now You can solve it without resorting to completing the box We can start with x ^ 2 – 4x = 5 We can subtract 5 from both sides and get x ^ 2 – 4x – 5 = 0 And you could say, if I have a -5 x Positive 1, so their product is -5 and their sum is -4 So I can say that this is (x – 5) (x + 1) = 0 So we’ll say x = 5 or x = -1 And in this case, perhaps this A faster way to solve the problem But the perfect thing about completing the square is that it is It always works It will always succeed regardless of transactions or No matter how difficult the issue is And let me prove this to you Let’s solve an apparent issue Difficult if we try to solve it by Factor analysis, especially if we solve it using aggregation or something like that Let’s assume we have a 10x ^ 2 – 30x – 8 = 0 Now, by the above, we can say, see Perhaps we can divide both parties by 2 This simplifies the equation slightly Let’s divide both sides by 2 If we divide everything by 2, by what do we get? We get 5x ^ 2 – 15x – 4 = 0 But again, we now have 5 in front of this Labs and we have to solve them using aggregation, which is It is a tired style But we can now go directly to completing the square, and even I do so I will divide both parties by 5 in order to get Parameter 1 is here You will see why this is different We did it traditionally If I divide everything by 5, I can I swear by 10 from the start, but I wanted to go to this Step up to show you this It did not help us much Let’s divide everything by 5 If we divide everything by 5, we get x ^ 2 – 3x – 4/5 = 0 So maybe you would say, why do we analyze the factors? By grouping? If we can always divide this parameter We can get rid of that We can always convert this to 1 or -1 if we divide On the right number But note, when we do this we’ll get 4/5 here So it’s very difficult to use factor analysis here You will have to say, what are the two numbers? Whose product is -4/5? It’s broken and when I take their sum, it is -3? It is an issue that is difficult to analyze It is hard to use factor analysis here The best thing to do is to use the checkbox So let’s think a little bit about how we can transform this Into an entire square What better to do – you will see the solution in several other ways I will show you all the methods because you will see that the teachers are using All roads – I’d prefer to place 4/5 on the other side So let’s add 4/5 to both sides of this equation You don’t have to follow this method, but I’d rather put the 4/5 Away Then what will we get if we add 4/5 To both sides of this equation? The left side of the equation becomes x ^ 2 – 3x, without having 4/5 here I will leave some space here This will equal 4/5 Now, as in the last issue, we’ll turn this around The left side into a binomial equation is square whole how to do that? Well, we say, what is the number we multiply by 2 and the sum is -3? The number of x 2 = -3 Or we take -3 and divide it by 2 This equals positive 3/2 Then we square positive 3/2 So in this example, we’ll say a = -3/2 And if we square 3/2, what do we get? We get positive 9/4 I took half of this lab, and square it, and we got Positive 9/4 The purpose of doing this is to turn the left side Into an entire square Now, we’re doing an aspect of the equation, on us To do to the other side So we added 9/4 here, so let’s add 9/4 there How does the equation become? We get x ^ 2 – 3x + 9/4 = – so let’s see if We can have a common denominator 4/5 equals 16/20 Multiply the numerator and denominator by 4 + / 20 9/4 equals multiplying Multiply by 5 to get 45/20 How much is the result of 16 + 45? As you can see, it becomes bifurcated, however I think it is fun Complete the box at some times 16 + 45 See this is equal to 55, 61 This is equal to 61/20 Let me rewrite it x ^ 2 + 3x + 9/4 = 61/20 Complex number Now this, at least on the left It is considered a complete square This is equivalent to (x – 3/2) ^ 2 It had a target -3/2 x -3/2 = positive 9/4 -3/2 + -3/2 = -3 So this squared is 61/20 We can take the square root of both sides and get x – 3/2 = + or – The square root of 61/20 Now we can add 2/3 to both sides of this equation And we get x = positive 3/2 + or – The square root of 61/20 This number is complicated and I hope you are clear You will not be able to – at least, I will not I be able to – get this number by analyzing the factors And if you want the current value, you can extract it Calculator You extract the calculator Then let me erase all this . 2/3 – Let us deal with the positive image first. So we want to We find 3 ÷ 2 + the second square root We’ll choose the yellow square root The square root of 61 ÷ 2, equal to 3.24 This is complicated 3.2464, I will write only 3.246 This equals approximately 3.246, and this was Positive image Let’s solve the minus image We can put the doorway – if it deals with the second and Next is the entry, so we want the entry to be yellow, for this Cause you pressed the second button So click the “Apply” button, you put it where you put it, we can We change the plural to subtract and We get -0.246 And you can make sure it checks The original equation The original equation was above here Let me check one of them Let’s suppose we have– So the second answer on the graph machine is The last answer that we used If you use a variable answer, which is that number Here If I have an answer box – I use the answer Which represents -0.24 Answer box -3 x Answer-4/5 – 4 ÷ 5- Equal – Here’s some clarification It doesn’t store the whole number, it turns into Somewhat rounded It stores a certain number of homes So when you compute it using this number stored here You get 1 x 10 ^ -14 So this is 0.0000 13 zero and then 1 Decimal point, followed by 13 zero and then 1 There are many zeros Or in fact, if you get the exact answer here, then You did an infinite approximation here, or Maybe if you keep this root picture, you’ll get This is really 0 I hope you find this helpful, meaning Complete the whole box Now we will expand it to the current quadratic formula That we can use, we can connect Things in order to solve the quadratic equation .

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