Intro to Quadratic Equations - Video Tutorials & Practice Problems

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Introduction to Quadratic Equations

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Hey ruin. We've learned to solve a couple of different types of equations. And now we're gonna add a new equation to the mix called a quadratic equation. Now, a quadratic equation is gonna be a bit more complicated than a linear equation. And it might even seem a bit overwhelming at times. But don't worry, I'm gonna walk you through everything you need to know about quadratic equations in the next several videos starting with what a quadratic equation even is and then going into how to solve them. So let's go ahead and get started. Now, if you take a linear equation, so like two X minus six equals zero and you simply add an X squared term. So if I had a three X squared plus two X minus six equals zero, this is now a quadratic equation. Now quadratic equation is also called a polynomial of degree two because the degree or power in my equation is this two on my X squared term. So whether you hear it called a quadratic equation or a second degree polynomial, these two mean the same thing. Now you're often going to have to write quadratic equations in standard form and the standard form of a quadratic equation is AX squared plus BX plus C equals zero where all of my terms are on the same side of my equation. And they're all written in descending order of power. So my very first term A X squared has a power of two where my second term BX, I can imagine an invisible one right here. So I have a power of one and then my last term is just a constant, there is no power. So I go from 2 to 1 to 0, descending order of power. Now you're not only going to have to write quadratic equations in standard form, but you're also gonna have to be able to identify each of those coefficients A B and my constant C. So looking at my example of here, this three X squared plus two X minus six, if I asked you what a in that equation was, I would be able to say that this three, since it's the coefficient of my X squared term is A, then B is the coefficient of my X term, which in this case is this positive two, this is B and then lastly C my constant term is just the constant on the end of my equation. So in this case, it is a negative six, I always wanna make sure that I'm looking at that sign so that I can completely identify my constant or any other coefficient correctly. So let's go ahead and take a look at some other quadratic equations. So I wanna write each of these in a standard form and then identify what A B and C are. So let's take a look at our first example. Here, we have five X squared equals X minus three. Now to get this in standard form, I want all of my terms on the same side. So I'm gonna go ahead and move this X over and then my negative three over. So they're all on that left side now to get rid of the X on my right side, I need to go ahead and subtract it. Now, remember whatever you do to one side of equation you have to do to the other that has not changed. And then I need to get rid of my three by adding it. So of course, cancel on that right side, leaving me on the left side with five X squared minus X plus three equals zero because there is nothing left on that right side. Now, I of course, want these to be in descending order of power. So I have two and then my, my invisible one here and then my three doesn't have a term. So I'm good and this is my answer. So this is my quadratic equation in standard form five X squared minus X plus three equals zero. So let's go ahead and identify A B and C in this equation. So A is going to be the coefficient of the first term of my X squared term. This is also called the leading coefficient because it's the very first coefficient in my quadratic equation. So A in this equation is five, then B is the coefficient of my X term. So here I have negative X which I have an invisible one here multiplying that X so my B is negative one, then lastly my constant term is this positive three on the end here. So C here is three. And that's all for that first example. So let's go ahead and look at one more. So over here, I have negative two X squared plus five thirds is equal to zero. Now, the first thing we want to do is get all of our terms to one side and they already are here. So we're good. And then we want to check that they're in descending order of power. Well, my first term has a power of two and then my second term is just a constant, there is no power. So they actually already are written in descending order of power. And actually this is already completely in standard form. So this is my quadratic equation in standard form. Let's go ahead and identify A B and C. Now A again is the coefficient of my X squared term. So in this case, I have a negative two multiplying that X squared. So that is A B is the coefficient of my X term. But if I take a look at my equation, I actually don't have an X term at all. So B is actually going to be zero because this would be like having a plus zero X stuck in that equation, which wouldn't do anything, which is why it's not there. And then C is my constant term, which in this case is actually a fraction and it is five thirds. So it's totally fine for any of our A B and C to be a fraction or for B and C to even be zero. As long as I have that X squared term, it is still a quadratic equation. So that's all for this video. And I'll see you in the next one.

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Problem

Problem

Write the given quadratic equation in standard form. Identify a, b, and c. $-4x^2+x=8$

A

a = - 4, b = 0, c = - 8

B

a = - 4, b = 1, c = 8

C

a = - 4, b = 1, c = - 8

D

a = 2, b = 1, c = 0

3

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Solving Quadratic Equations by Factoring

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6m

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Hey, everyone, whenever we solved linear equations, we wanted to find some value for X that we could plug back into our original equation to make it true. And we want to do the same thing when solving quadratic equations, we want to find some value for X that makes our equation true. But whenever we solve linear equations, we were able to move some numbers around and then isolate X to get an answer. But what happens if I try to do the same thing with the quadratic equation while looking at this X squared minus five X plus four over here, I might go ahead and move this five X to the other side, maybe move my four as well. Leaving me with X squared is equal to five X minus four. And then you might think, oh, we can just take the square root to get X by itself which would leave me with X is equal to where do I go from there? What is the square root of five X minus four? So I can't actually solve a quadratic equation the same way solve a linear equation. And more than that, there are actually often going to be two correct values for X when solving a quadratic equation. So how do we find those two values? Well, luckily this something else that we need is something that we already know how to do from the previous chapter factoring. So let's go ahead and take a look at how we can factor to solve quadratic equations. Well, we're going to want to factor from standard form and then, and simply set each factor equal to zero. So if I have a quadratic equation like X squared plus X minus six is equal to zero, and then I go ahead and factor that into X plus three times X minus two equals zero. I can simply take each of those individual factors. So X plus three and X minus two, set them equal to zero and then solve for X. And the reason that I can do this is because these factors are multiplied together. So X plus three times X minus two is going to equal zero and anything times zero is zero. So if either one of my factors, let's say that X plus three was +00. If I multiply that times anything that would give me zero, so that might seem a little bit abstract right now. Let's go ahead and take a look at that in action. So in my example, here I want to solve this by factoring and I have X squared minus nine, X is equal to negative 20 So our first step here is going to be to write our equation in standard form. So I want to go ahead and get all my terms on the same side in descending order of power. So to do that, I just need to move my negative 20 to the other side, which I can do by adding 20 to both sides, it will cancel on that right side, leaving me with X squared minus nine X plus 20 is equal to zero. And we're done with step one. Now, step two is going to be to factor completely. Now remember there are multiple different methods to factor. So let's go ahead and take a look at what method of factoring we should use for this equation. So we first want to look at how many terms this has and it has this X squared negative nine X and positive 20. So that is three terms which tells me that I need to then say does it fit a factoring formula? And looking quickly at my factoring formulas here, it doesn't seem to fit a factoring formula either. So my answer is no, which means that I then need to use the AC method to factor this quadratic. So let's go ahead and use the AC method here. So first we want to look at A and C. So let's go ahead and make sure that we know what A B and C are here. So in this case, I have this kind of invisible one. And that is my A, then B is negative nine and C is 20. So if I multiply A and C together A is just one. So it's really just C so AC is 20. So I want to find two factors that multiply to 20 then add to B which in this case is negative nine. So two factors that multiply to 20 add to negative nine. Since that B is negative, I know that both of my factors have to be negative. So let's think of factors of 20 that are negative. So I have negative one and negative 20 then I have negative two and negative 10 and then I have negative four and negative five. Now negative one and negative 20 are going to add to negative 21 which is not negative nine, negative two and negative 10 are going to add to negative 12 which is also not negative nine. But then I have negative four and negative five which do add to negative nine. So it turns out that negative four and negative five are my factors here. So once I have figured out what my factors are, I can go ahead and write this as X minus four times X minus five is equal to zero and step two is done. Now for step three, I want to set my factors equal to zero and then solve for X. So let's go ahead and look at each individual factor. So I have X minus four. I'm setting that equal to zero and then X minus five equals zero as well. So let's go ahead and solve here to solve for X with this X minus four, I just need to add four to both sides. Leaving me with X is equal to four. And then with X minus five, I simply add five to both sides and I end up with X equals five. And these are actually my solutions. I have completed step number three. My solutions are X equals four and X equals five. Now, because we wanted to find two values for X that we could plug back into our equation. That makes it true. Let's go ahead and make sure that that does happen. So let's first try this X equals four and go ahead and plug that in. So if I plug it into my original equation, I will get four squared minus nine times four is equal to negative 20. Now, four squared is 16, 9 times four is 36. So 16 minus 36 is to negative 2016, minus 36 is negative 20 that's equal to negative 20. So negative 20 equals negative 20. That's definitely a true statement. I know that four is a value for X. That makes it true. It's definitely one of my answers. Now, we can do the same thing for five here, but I'm going to leave that up to you. So that is all we need to do in order to solve quadratic equations by factoring. Let's go ahead and get some practice.

4

Problem

Problem

Solve the given quadratic equation by factoring. $3x^2+12x=0$

A

$x=3,x=4$

B

$x=0,x=-4$

C

$x=-3,x=-4$

D

$x=1,x=4$

5

Problem

Problem

Solve the given equation by factoring. $2x^2+7x+6=0$