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Quadratic Expression Factoring

May 11, - FREE Factoring Quadratics Activity (Crazy Words - fun like Mad Libs!). In this activity, students factor quadratic expressions with a leading. Factoring quadratic expressions, beschreiben Worten essays selbst sich Sie in. Die Beschreibung von Quadratic Expression Factoring. Finding factors in quadratic expressions is a very important unit in general mathematics. It is also very.

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Quadratic Expression Factoring

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Quadratic Expression Factoring Factor quadratic equations step-by-step Video

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I think the more examples we do the more sense this'll make. Let's say we had x squared plus 10x, plus-- well, I already did 10x, let's do a different number-- x squared plus 15x, plus And we want to factor this.

Well, same drill. We have an x squared term. We have a first degree term. This right here should be the sum of two numbers.

And then this term, the constant term right here, should be the product of two numbers. So we need to think of two numbers that, when I multiply them I get 50, and when I add them, I get And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally.

So what could a and b be? Let's think about the factors of It could be 1 times Let's see, 4 doesn't go into It could be 5 times I think that's all of them.

Let's try out these numbers, and see if any of these add up to So 1 plus 50 does not add up to But 5 plus 10 does add up to So this could be 5 plus 10, and this could be 5 times So if we were to factor this, this would be equal to x plus 5, times x plus And multiply it out.

I encourage you to multiply this out, and see that this is indeed x squared plus 15x, plus In fact, let's do it.

Notice, the 5 times 10 gave us the The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x, plus Let's up the stakes a little bit, introduce some negative signs in here.

Let's say I had x squared minus 11x, plus Now, it's the exact same principle. I need to think of two numbers, that when I add them, need to be equal to negative And a times b need to be equal to Now, there's something for you to think about.

When I multiply both of these numbers, I'm getting a positive number. I'm getting a That means that both of these need to be positive, or both of these need to be negative.

That's the only way I'm going to get a positive number here. Now, if when I add them, I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number, so the fact that their sum is negative, and the fact that their product is positive, tells me that both a and b are negative.

Remember, one can't be negative and the other one can't be positive, because the product would be negative.

And they both can't be positive, because when you add them it would get you a positive number. So let's just think about what a and b can be. So two negative numbers.

So let's think about the factors of And we'll kind of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6.

Now, which of these when I multiply these-- well, obviously when I multiply 1 times 24, I get When I get 2 times sorry, this is 2 times I get So we know that all these, the products are But which two of these, which two factors, when I add them, should I get 11?

And then we could say, let's take the negative of both of those. So when you look at these, 3 and 8 jump out. But that doesn't quite work out, right?

Because we have a negative 11 here. But what if we did negative 3 and negative 8? Negative 3 times negative 8 is equal to positive Negative 3 plus negative 8 is equal to negative So negative 3 and negative 8 work.

So if we factor this, x squared minus 11x, plus 24 is going to be equal to x minus 3, times x minus 8. Let's do another one like that. Actually, let's mix it up a little bit.

Let's say I had x squared plus 5x, minus So here we have a different situation. The product of my two numbers is negative, right?

My product is negative. That tells me that one of them is positive, and one of them is negative. And when I add the two, a plus b, it'd be equal to 5.

And what combinations of them, when I add them, if one is positive and one is negative, or I'm really kind of taking the difference of the two, do I get 5?

So if I take 1 and I'm just going to try out things-- 1 and 14, negative 1 plus 14 is negative Negative 1 plus 14 is So let me write all of the combinations that I could do.

And eventually your brain will just zone in on it. So you've got negative 1 plus 14 is equal to And 1 plus negative 14 is equal to negative So those don't work.

That doesn't equal 5. Now what about 2 and 7? If I do negative let me do this in a different color-- if I do negative 2 plus 7, that is equal to 5.

We're done! That worked! I mean, we could have tried 2 plus negative 7, but that'd be equal to negative 5, so that wouldn't have worked.

But negative 2 plus 7 works. And negative 2 times 7 is negative So there we have it. We know it's x minus 2, times x plus 7. That's pretty neat.

Negative 2 times 7 is negative Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well honed this skill.

So let's say we have x squared minus x, minus So the product of the two numbers have to be minus 56, have to be negative And their difference, because one is going to be positive, and one is going to be negative, right?

Their difference has to be negative 1. And the numbers that immediately jump out in my brain-- and I don't know if they jump out in your brain, we just learned this in the times tables-- 56 is 8 times 7.

I mean, there's other numbers. It's also 28 times 2. It's all sorts of things. But 8 times 7 really jumped out into my brain, because they're very close to each other.

And we need numbers that are very close to each other. And one of these has to be positive, and one of these has to be negative.

Now, the fact that when their sum is negative, tells me that the larger of these two should probably be negative. So if we take negative 8 times 7, that's equal to negative And then if we take negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there.

So when I factor this, this is going to be x minus 8, times x plus 7. This is often one of the hardest concepts people learn in algebra, because it is a bit of an art.

You have to look at all of the factors here, play with the positive and negative signs, see which of those factors when one is positive, one is negative, add up to the coefficient on the x term.

But as you do more and more practice, you'll see that it'll become a bit of second nature. It is pretty strait forward if you follow all the Solving quadratics by factorizing link to previous post usually works just fine.

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