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### Solving Quadratic Equations by Factoring

- A quadratic equation is an equation where the largest power present on a variable is two.
- Factoring can help solve quadratic equations because of the zero factor property. If the quadratic is set equal to zero, and then factored, then each factor can be set equal to zero and solved separately.
- Any of the previous factoring techniques for trinomials can be used such as reverse FOIL or the AC method.

### Solving Quadratic Equations by Factoring

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:08
- Solving Quadratic Equations by Factoring 0:19
- Quadratic Equations
- Zero Factor Property
- Zero Factor Property Example
- Example 1 4:00
- Solving Quadratic Equations by Factoring Cont. 5:54
- Example 2 7:28
- Example 3 11:09
- Example 4 14:22
- Solving Quadratic Equations by Factoring Cont. 18:17
- Higher Degree Polynomial Equations
- Example 5 20:22

### Algebra 1 Online Course

### Transcription: Solving Quadratic Equations by Factoring

*Welcome back to www.educator.com.*0000

*In this lesson we are going to take a look at how you can solve quadratic equations using the factoring process.*0002

*Specifically we will focus on that factoring process and look at how factoring can help us to solve a few other types of equations.*0010

*Recall that when I'm talking about a quadratic equation and talking about any equation of the form ax ^{2} + bx + c = 0.*0022

*We do not want our a to be a 0 value.*0032

*We do not want to get rid of that x ^{2} term.*0036

*Some examples of quadratic equations are below x ^{2} + 5x +6 that is definitely a quadratic equation.*0040

*You can see that it fits the form pretty good.*0048

*A would be 1, b would be 5 and c would be 6.*0051

*It also applies to other types of equations like this one.*0055

*Notice how this does not quite look like the form,*0059

*but if you manipulate it just little bit and move the 3 to the other side it actually does fit the form quite well.*0061

*One key feature of these quadratic is that you will have your squared variable in there somewhere.*0071

*Even this last one is a good example of a quadratic equation.*0078

*This one now you can move the 4 to the other side, and put in a 0 placeholder to show that it does fit the form of our quadratic equation.*0084

*Keep looking for that squared term to be able to hunt these out.*0093

*The way we are going to solve these quadratic equations is we are going to use a very special property known as the 0 factor property.*0101

*What the property says is that if you have two numbers call them a and b.*0109

*If those numbers multiply together and get you 0 in, either a or b must be 0.*0114

*The way I like to think of this is much in the context of a game.*0121

*Pretend that I have two numbers in my head, and they multiply together and get 0.*0125

*If I would ask you what you know about these numbers you would probably tell me that one of them better be a 0.*0131

*Because that is the only way we are going to multiply it and get that 0.*0135

*That is exactly what we are trying to say.*0139

*If the product of two numbers is 0, then one of the numbers or maybe both of them are 0.*0142

*Watch how we use the 0 factor property because they get to our solutions for these quadratics.*0147

*We are going to use that to solve something like x +4 × x - 5 = 0.*0157

*The reason why the 0 factor property is going to come in to play is because we treat each of these factors like one of those numbers.*0163

*If I'm multiplying them together and I get 0 then I know that one of these factors or may be both of them are equal to 0.*0173

*Let us take these factors and create new equations for them.*0181

*I know that x + 4 = 0 or x – 5 = 0.*0188

*The reason why that helps us out is notice how these new equations we form down here, they are much simpler than the original.*0197

*In fact, they are nice linear equations and I can easily solve them just by subtracting 4 from both sides of this one.*0205

*I can solve the other one by simply adding 5 to both sides.*0217

*What that 0 factor property helps us to do is take each of the factors that are in our quadratic and set them both equal to 0.*0226

*That way we have some much simpler equations that we can solve from there.*0236

*Let us try some other ones that is already been factored out.*0243

*I want to solve 2x + 3 × 5x + 7 = 0.*0245

*We will take each of the factors here and we will set them equal to 0 individually.*0251

*This one is equal to 0 and 5x + 7 will make that one equal to 0 as well.*0258

*It is just a matter of just solving these individually.*0265

*We will subtract 3 from both sides and we will divide by 2.*0268

*I know that one of my solutions is x = -3/2.*0279

*Let us solve the other one, -7 – 7 that will give us 5x = -7, divide both sides by 5 that would give us x = -7/5.*0284

*We have two solutions.*0307

*The 0 factor property works out good your factors are not that big.*0312

*In this one I have an x × x + 6.*0317

*We will take each of these factors and we will set them equal to 0.*0323

*There is not a whole lot of solving to be done with it each of these.*0332

*In fact, one of them are already solved, x could be 0.*0334

*On the other one, I will just subtract 6 on both sides, x = -6.*0339

*I have both my solutions for that one.*0350

*We rarely have a quadratic equation that is already been factored out for us.*0356

*In fact, these first few examples here are very nice and simple ones to solve since they were already factored.*0361

*Our goal is to take any type of quadratic equation and factor it on our own using many of those different factoring techniques that we have learned.*0367

*Once we do have it factored then we will use our 0 factor property on it so that we can find our solutions.*0377

*Here is a good outline of what that might look like.*0383

*Maybe I will start with equation like x ^{2} - 3x -10=0.*0386

*I could use reverse foil or the AC method that actually factor that polynomial.*0391

*Once I have my factors I could set them each equal to 0 and solve and then get my solutions.*0397

*It is a good idea to just check your solutions by substituting them back into the original.*0403

*If you do this, you want to put it in for all copies of that variable.*0408

*If I'm checking to see if 5 is a solution, I will put in for my x that has been squared.*0414

*I will put that in for the x right next to the 3 just so I can make sure that it works out in the entire equation.*0420

*I get 25 – 15 – 10, does that equal 0?*0429

*Sure enough, it does because of 25 – 25 that is how I would know that a solution checks out.*0437

*If we are going to do our factoring properly it means we always have to make sure that it set equal to 0 first.*0453

*Make sure that it is one of your first few steps when working on quadratic equation.*0458

*Let us give these two a try.*0463

*The first one is x ^{2} + 2x = 8.*0465

*I know it is one of my quadratics, I can see my squared term right there.*0469

*What I’m going to do is I'm going to get it equal to 0, x ^{2} + 2x.*0474

*Let us go ahead and subtract 8 from both sides -8 = 0.*0481

*Let us use our factoring techniques to see if we can break it down.*0490

*This one was not that big, I'm going to use just the reverse foil method to see if I can figure out what is going on here.*0495

*Two numbers that will multiply and give me x ^{2}, that better be an x and another x.*0504

*I need two things that will multiply to give me -8 but somehow add to give me 2.*0513

*4 and 2 will work as long as my 4 is positive and my 2 is negative.*0520

*I have both of my factors x + 4 and x - 2.*0529

*I will take each of these and set them equal to 0.*0535

*Solving this one over here I will subtract 4 and solving the other one I would add 2.*0545

*Giving me the solutions x = -4 and x = 2.*0553

*You may have already notice that you can shortcut that last few steps just a little bit.*0566

*This number here will always be the opposite of this one right here in the factor.*0572

*I recommend going and least showing those steps for the first 3 times*0579

*so you can be assured that you are using the 0 factor property in the background.*0584

*If you want to get a little bit more familiar with it, then feel free to use that short cut.*0588

*The next one is x ^{2} = x + 30.*0595

*We will start this one off by getting everything over to one side.*0599

*I'm subtracting the x over and subtracting the 30 now it is equal to 0.*0603

*This was not that big either, so let us try reverse foil on that.*0611

*Two terms that would multiply to give me x ^{2} would have to be x and x.*0616

*And two things that would multiply to give me 30, but add to be -1, I think we are looking at -6 and 5.*0620

*That will definitely do it.*0632

*We can take each of these and set them equal to 0.*0636

*Solving them I might have to add 6 to both sides of this one and subtract 5 from both sides of this one.*0646

*Leaving me with two solutions that x = 6 and x = -5.*0652

*Remember that if you are ever unsure of these answers here*0660

*feel free to put them back into the original just to make sure that they do work out.*0664

*Let us try some more that are just a little bit more complicated.*0672

*This next one is 3m ^{2} – 9m = 30.*0676

*I’m going to get it equal to 0 first.*0681

*That looks pretty good.*0692

*Now I'm going to try and factor it.*0694

*One of the very first things I like to check for factoring is they all have something in common.*0697

*Unfortunately, it looks like these ones do.*0703

* Everything in here is divisible by 3.*0705

* We will take out that common 3 before we get too far, that way we can make our factoring process much easier.*0708

* m ^{2} is the left 9 ÷ 3 = 3m and 30 ÷ 3 =-10.*0716

*We just have to factor this.*0725

*m ^{2} – 3m -10.*0727

*Two numbers that would multiply and give me m ^{2} would have to be m and m.*0737

*Two numbers that would multiply to give me -10, but add to be -3, that will be - 5 and 2.*0746

*We have our factors, it is time to take all of them and set them equal to 0.*0758

*3 is that equal to 0? m -5 does that equal 0? and m + 2 is that equal 0?*0764

*Notice how I even took that 3, one common mistake is just to say that it is one of your solutions,*0773

*but from 0 factor property you are checking to see if it equal 0.*0779

*Notice how 3 is not equal to 0, that does not make any sense.*0784

*We will not even consider that.*0787

*The other ones I can go ahead and continue solving.*0790

*Add 5, add 5 and – 2m and -2.*0793

*m = -5 and m = -2.*0799

*I have my solutions for this quadratic equation.*0805

*By looking for that greatest common factory you can often save yourself quite a bit of work.*0811

*In this one I’m going to get my 3x to the other side.*0816

*Then I immediately noticed that both of these have an x in common, so I can actually pull that x out.*0822

*I have an x and x -3, those are my two factors that I will set equal to 0.*0834

*It is nice having one of our factors equal to 0 already it means that we do not have to do much of the solving process.*0841

*It is already done.*0847

*I will go ahead and add 3 to the other equation that we formed that way that one is solved.*0850

*x = 3.*0855

*We have both of our solutions for that one.*0857

*A tricky part of this entire thing is to make sure that it is set equal to 0 first.*0864

*Sometimes you might have to do some simplifying or combine things together before you can set it equal to 0.*0869

*In this one we have x × 4x + 7 and all of that is equal to 2.*0876

*It is not that difficult to get it equal to 0 since we would simply subtract 2 from both sides.*0882

*But in order to have factors I need everything to be multiplied together and I still have some subtraction in here.*0892

*What I'm going to do is distribute with my x here and then I will go ahead and try factoring.*0899

*4x ^{2} + 7x – 2 = 0.*0906

*This one is a little bit more complicated, it is not very obvious what the reverse foil method should do on it.*0916

*Let us go ahead and dig up our AC method and give that a try.*0924

*The AC method we will multiply a and c together 4 × -2 = -8.*0929

*I'm looking for two numbers that will multiply to give me -8 but add to be 7.*0939

*What do we have for possibilities, it could be 1, 8, it could be 2 and 4.*0946

*It could be any of those in reverse.*0951

*We want them to multiply to give us -8 and we want it to add to be 7.*0955

*It looks like I have to use that 1 and 8.*0965

*The way this will work out as it will be -1 and 8.*0971

*That will help me split up my middle term.*0976

*We have done that we can go ahead and continue using factor by grouping and I will take out a common 8 from these first two terms.*0993

*The next two, I can see a common 2 that I can pull out.*1010

*And now that we have that I can say what my factors are.*1027

*We have 4x - 1 and x + 2.*1035

*That is quite a lengthy process but recognize we are not done yet.*1042

*All that was just done to factor it.*1044

*We need to take each of the factors and set them equal to 0.*1047

*4x - 1 = 0 and x + 2 = 0.*1053

*We will add one to both sides and divide by 4.*1062

*We have one of our solutions that x could equal ¼.*1071

*The other one let us go ahead and subtract 2.*1078

*Our other solutions is x =-2.*1083

*The factoring process can be lengthy, but it is just one step along the way to finding the solutions.*1090

*Let us try another one.*1099

*Some higher degree polynomials can also be solved using this factoring process.*1101

*Usually the ones that come to mind are the ones that have a greatest common factor that you can go ahead and dry out first.*1106

*Notice how this one is not a quadratic, it actually has x ^{3} in there.*1112

*I can still use some of my factoring techniques because they both have 2x in common.*1118

*Let us pullout that 2x.*1126

*I would have x ^{2} - 25.*1130

*It looks like that what is left over happens to be a quadratic, but it is actually even more special than that.*1138

*This right here is the difference of squares which means I can use one of my short cut formulas to help me out.*1146

*This would be x + 5 and x – 5.*1162

*I was able to factor it down completely and now you can take all of the pieces and set each one to 0.*1169

*2x, that could equal to 0, x + 5 that could equal 0, and x - 5 that could equal 0 as well.*1178

*Divide both sides by 2 for this one.*1191

*That will give us x could be 0.*1195

*-5 on this one x could equal -5, add 5 to both sides of this one, x= 5.*1200

*This one has 3 solutions that I could go back and check.*1214

*Let us try out everything we have learned with factoring to see if we can tackle out one more problem.*1225

*This one is quite large.*1229

*This one involves x -1 × 2x – 1 = x +1 ^{2}.*1231

*It looks like it might be quadratic after I do have something squared in there, but I have to get it equal to 0 first.*1239

*I need to factor it from there and I actually have to do some multiplying before actually get to that factoring process.*1248

*We will start over here on the left side.*1257

*Let us go ahead and do some foiling okay.*1258

*My first terms would be 2x ^{2}, outside –x, inside - 2x, and my last terms 1.*1262

*Over on the right side of this, this is the same as x +1 × x +1.*1281

*We are going to use foil to help us spread this out.*1289

*Our first terms on that side x × x = x ^{2}, outside terms x, inside terms x, and last terms 1×1 =1.*1293

*I have better freedom on combining things together.*1306

*Let us combine together these x's and these x's.*1310

*2x ^{2} - 3x + 1= x^{2} + 2x + 1.*1318

*You can see that this is definitely quadratic.*1332

*I got lots of x ^{2} and I have a much better task of getting it all to one side and getting it equal to 0.*1335

*I'm going to subtract the x ^{2} from both sides.*1343

*That will combine those ones and subtract 2x and we will subtract a 1.*1346

*This will give us x ^{2} - 5x = 0.*1356

*This seems like a much nicer problem to solve.*1364

*It even has a greatest common factor of x that I can take out from both of these terms.*1366

*I can take each of these factors and set them equal to 0.*1378

*Looks like one of those are already solved for us, so we will just leave that as it is.*1389

*The other one we will add 5 to both sides and then we will get our second solution that x must equal 5.*1393

*Some quadratics you can factor them using many of the factoring techniques that you learn before.*1405

*Take each of those factors and set them equal to 0 so you can find your solutions.*1411

*Thank you for watching www.educator.com.*1416

1 answer

Last reply by: Professor Eric Smith

Wed Jul 16, 2014 8:12 PM

Post by patrick guerin on July 15, 2014

Thanks for the lecture!

1 answer

Last reply by: Professor Eric Smith

Wed Jul 16, 2014 8:12 PM

Post by patrick guerin on July 15, 2014

On example 3(3m2-9m=30), when you got to the zero factor property, could you possibly have it where you have 3(m+2) and 3(m+5) also with that problem?

1 answer

Last reply by: Professor Eric Smith

Wed Jul 16, 2014 8:04 PM

Post by patrick guerin on July 15, 2014

You said that some equations can be changed into the quadratic form. How would you know easily if they can be changed into a quadratic equation or not?

1 answer

Last reply by: Professor Eric Smith

Wed Apr 30, 2014 6:30 PM

Post by Fahad Chandia on April 30, 2014

In Example 5 you subtract x square from 2x square ,So reminder should b 2 not x Square .Can you explain to me ?

0 answers

Post by Fahad Chandia on April 30, 2014

if you put minus x 2