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\nSo what I'd like to do is show you how to solve a quadratic equation and by completing the square. Now, typically, completing the square, we look into factoring. And if it's not factorable, we can use the quadratic formula. Now, a...\nSo what I'd like to do is show you how to solve a quadratic equation and by completing the square. Now, typically, completing the square, we look into factoring. And if it's not factorable, we can use the quadratic formula.
Now, a quadratic formula is great, and I would probably prefer that over completing the square. However, the process of completing the square is very, very important, especially for other topics. And you can also complete the square to solve a quadratic equation, so that's basically what we're going to go ahead and practice.
Now, the act of completing the square. Basically, what we're going to do is create a perfect square trinomial. Once we have a perfect square trinomial, we can rewrite that as a binomial squared, which then allows us to use our inverse operations-- the square root method-- to solve for x.
So when we can't factor something-- and you can see, in all these examples, these would all be examples of quadratics we cannot factor. What we can do is rewrite it. Rather than plugging it into some formula and figuring out what it is, we can also complete the square. Rewrite it in a different format, so, therefore, we can solve without having to factor.
So basically, without further ado, let's go ahead and get started on what exactly I'm talking about. So basically, what we're going to do is we're going to group two terms. And what we're going to do is we're going to create a perfect square trinomial out of that.
Now, remember, a perfect square trinomial is a trinomial where the first term and the last term are square terms. So what we need to do is we need-- if I'm going to group these first two terms-- now, remember, whatever you're grouping your terms, you've got to make sure that your quadratic term has a coefficient of 1, which this does.
Now, inside these parentheses, we need to create a term that is going to make this a purpose square trinomial. So if x squared is my first term, which is a square number, 4x is my middle term. To find our term that's going to complete the square, basically, what we're going to do is take b, divide it by 2, and square it.
Now, remember, you might say, well, where does b come from? Well, remember, ax squared plus bx plus c. So b is going to be the coefficient of our middle term. So what I have is 4 divided by 2 squared. 4 divided by 2 is 2. 2 squared is equal to 4.
So now, I have x squared plus 4x, and then I'm going to add that number plus 4 inside the parentheses. Now, you just can't randomly add numbers inside of here. Basically, what you need to do is, whenever you add a number on one side of the equation, you have to add it on the other side of the question or add and subtract it on the same side of the equation. Now, since we're solving, I'm going to go ahead and add it to the other side of the equation.
OK. So we add it on the left side, and we add it on the right side. Now, what I have done is I have created a perfect square trinomial, and what's nice about creating a perfect square trinomial is we can always factor them down to a binomial square.
So I'd have x plus 2 squared because I factored that perfect square trinomial. Then I can add the 14 to the other side. Let's use a red.
So therefore, I have x plus 2 squared is equal to 28. Now, to go ahead and solve, I'm just going to use my inverse operations. So I'll take the square root of both sides, and I have x plus 2 is equal to plus or minus the square root of 28.
Now, to solve for x, I subtract 2 on both sides and then we'll also want to simplify the square root of 28. And so we want to see what square numbers divide into 28. And let's see, I have 4. 4 goes into there 7 times, so my final answer is x equals plus or minus negative 2 plus or minus.
I can break this down. Let's see. The square root of 28. I can break that down into 4 times 7.
The square root of 4 is 2 times and square root of 7. So 2 square root of 7. And there you go.
All right. So in this example, we're going to the same thing. Another thing, actually, I like to do-- which I kind of forgot about-- is, since we're always going to be completing the square with the first two numbers, let's just get rid of this 7. We don't really need it right now on this left side. So I'm going to add 7 to the right side, and that's going to now leave me with x squared minus 10x equals 7.
Now, I'll group my first two terms, and I'll say, all right. Now, I need to complete the square. I need to create and make this a perfect square trinomial. So therefore, I can factor it.
So again, I just do negative 10 divided by 2 and square it, and that equals 25. Negative 10 divided by 2 is going to be negative 5. Negative 5 squared is 25.\n
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\n- Grade
- 12th - Higher Ed
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\n- Subjects
- Math
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\n- Source:
\n\nBrian McLogan\n\n
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\n\n- Language
- English
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