We're going to start this new unit by looking at a quadratic equation. A quadratic equation has an x squared term in it. That's where the quadratic comes in. Let's look at one student's attempt to try and solve this. The student attempted to solve for x by isolating the variable. First, the student subtracted x squared from both sides. And then subtracted 6 from both sides. Finally to isolate x, the student divide both sides by 5. So did the student find the number that x equals? Yes or no?
Well, no. The student has not found a number for x. The student did manipulate or change the equation up here, which is great, but we still don't know the value of x. You might be able to look at the original equation and do some mental math to come up with the values for x that make the equation true, but what we're going to do in the upcoming lessons is figure out a solving method for this exact kind of problem.
We're going to keep coming back to this equation as we learn new skills to see if we can solve it. But for now let's start by solving a number puzzle. For number puzzles, we're going to find 2 numbers that multiply to give us the top number and that add to give the bottom number. So for this one I want to find factors of 20 that sum to 9. We can think about it in our heads or make a list of the factor pairs. This is usually more helpful. We start with 1 since 1 divides into any number. 1 times 20 is 20. So this is definitely a factor pair. factor pair. I don't need to continue writing any more factor pairs because well I've repeated one of them already. Ten and two would end up being repeated as would 20 and one if I continued listing factor pairs so if I'm going in order I can just stop when I have a repeat. I look at my list and I see which one of the pairs adds up to nine, that must be this one. So four and five would be the answer to this number puzzle.
Try solving these two number puzzles. You want two factors that multiply to give you the top number, like 4 times 5 equals 20, and the two numbers need to add to the bottom number, 4 plus 5 equals 9. And it doesn't matter which order you put the numbers in. I could have also written five here and four here. You'll be correct either way.
To find the numbers here, we list the factor pairs of 24. 1 and 24 is a factor pair, since they multiply to give us 24. After 1, I should try 2. Well, 24 is even, so 2 definitely goes into it. 2 times 12 is 24. 3 goes into 24 as well, so doesn't go into 24, because, since this number doesn't end in a 0 or a 5. And then, if i try 6, I repeat a factor pair, so really, I'm done here. Any higher factor pairs will be repeated like 8 and 3, 12 and 2, and 24 and 1, so I look at my list of factor pairs and I see which pair adds to 10. Well, 4 and 6 add to that I've only listed the positive factors here. We could also have negative 1 and negative 36, but we'll talk about that later. I just want to find the pair that adds to 13. 4 and 9 add to 13, so that must be my answers for my number puzzle. Great work if you got those two right.
Throughout the next few lessons, we'll be talking about factors and terms. We want to make sure we know the meaning of each word, so let's start by focusing on the differences between their meanings. Terms are separated by addition or subtraction in an expression. So this expression has three terms, 3 xy, 4x and These are the things we multiply together to give us our term. The factors of 4x are 4 and x and the factors of 5 are 5 and 1. We could also write 4 time x as 2 times 2 times x. This would be the prime factors of this term. Prime numbers have only 2 distinct factors. Here's another case where we have 2 factors, 2x plus 5 is 1 factor and 7 is another. Within this 2x plus 5 factor, we have 2 terms. Positive 2x and positive 5.
Let's check our knowledge with a little check. Are these expressions a term or a factor? Choose the appropriate choice.
We know the 7 is multiplied by 4x plus 5y, so these two are factors. The 2 is separated from this multiplication by addition, so this is just a term. Within this factor of 4x plus 5y, we have two terms, 4x and positive 5y. So, this 4x must be a term as well. And finally, we know x and y are both factors since x appears in the multiplication for 4x, and y appears in the multiplication for
When we factor a number, we split up the number into the product of other numbers. 216 can be written as 2 times 108. This number is even, so I know I can divide by 2. Then I just continue my process to get all the other factors. 108 breaks down into 2 times 54, and 54 breaks down into 2 times 27. 27 isn't divisible by 2, but it is divisible by 3. So we have 3 times 9 is 27 and then 9 is 3 times 3. These factors are the prime factors, because the numbers only have two distinct factors, 1 and itself. The only factor of 2 is 2 and 1, and the only factor of 3 is 3 and 1. We can rewrite 216 using bases and exponents. We know we have 2 to the third power, since we have 3 2's. And we have three to the third power since we have three 3s. This is the prime factorization for 216, since the only numbers that appear here are prime factors. Use what you know about prime factors to figure out what's the prime factorization for 192. You can write your answer in this box and be sure to include the appropriate bases and exponents.
factor tree for a 192. Notice each time we're dividing our numbers by two. We have two to the six and one, three. So two to the six times three would be the prime factorization.
We're going to use these prime factorizations to find the greatest common factor for 216 and 192. The greatest common factor for these two terms, is the product of the lowest power of each of the factors that appears in each term. For example, if we looked at 12, and 44, we could see that they both share a two to the second power. They don't share a power of three, and they don't share a power of 11. So two squared, or four, would be the greatest common factor for 12 and 44. So what do you think is the greatest common factor for these two numbers? Write your answer in this box. You'll also want to simplify your answer. For example, if you think two cubed times five is the greatest common factor, you would really write that as 40, since two cubed is eight, and eight times five is 40.
The greatest common factor for these two numbers would be 24. Excellent work if you got this one correct. If we expand the prime factors of 216 and the prime factors of 192, we can see that they share three, two's and one, three. The greatest common factor is the product of the lowest power of each factor that appears in each term. So two to the third is the lowest power of the basic two that appears in both terms. And three to the one is also the lowest power that appears for the base of three in each term. We want to simplify our result so two cubed is eight and eight times three is 24.
The greatest common factor of variable expressions works just the same as it does with numbers. The greatest common factor or GCF is simply the product of the lowest power of each number factor and variable factor that appears in every term. So what do you think would be the greatest common factor for 45 x squared y cubed and 36 x to the fourth y squared z?
The GCF is 9 x squared y squared. Let's see why. This would be the prime factorization for 45 x squared y cubed. And this would be the prime factorization for 36 x to the fourth y squared z. I'm going to regroup my numbers together so that we have the 2's together and the 3's together. I want to find the lowest power of each factor that appears in both terms. We know that So that's going to start the product of my greatest common factor. This number has a 5, however this number does not, so I can't list 5 as part of my greatest common factor. However X is a factor that appears in both terms and we have a power of X squared that is common between them. So we can list X squared in the product of our GCF. And finally, they both share the lowest power of y squared, so we can include that in our GCF as well. The greatest common factor is the product of all of these factors together. So we have 9 times x squared times y squared.
We can actually use the greatest common factor to factor expressions like this. This would be factored form. We have one factor multiplied by another factor of two terms. When we factor a number out of other numbers, we want to think about what would be left, or what's remaining. For example, if we factor out 9 x squared y squared from our first term, we would be left with 5 times y. These are the left over factors remaining. This should also make sense, because if we multiply 9 x squared y squared by 5 y, we would wind up with 45 x squared y cubed. What do you think should go inside this box? What's our second term inside of our parenthesis?
Great work if you got 4x squared z. We can see that 2 times 2, times x times x times z, would be our leftover factors, so that makes 4x squared z. You might have also thought about it in your head. If you divide 36 by 9, you'll get 4. And if you divide x to the 4th, by x squared, you'll get x squared. If you divide y squared by y squared, you would be left with 1. And then you have one z remaining, so 4x squared z.
When we factor an expression like this, we're finding two numbers or expressions that multiply together to give us what we originally started with. We can quickly check by using multiplication. If I multiply 9x squared y squared by 5y and by 4x squared z. I should wind up with my original statement. This is where the properties of exponents come in handy. 9x squared y squared times 5y would be 45x squared y cubed. We add the exponents here. There's a 2 and a 1. And 9x squared y squared times 4x squared z would be 30 x to the 4th y squared z. 9 times 4 is 36, x squared times x squared is x to the 4th, and then we have y squared and z on the end.
Before we get to the practice for this lesson, let's try some number puzzles. This time the top numbers are negative. This means that one of that factors has to be negative, since a negative times a positive is a negative number. I want you to find the factors for these number puzzles. Remember, we want to find two numbers that multiply to the top number, but some to the bottom number, And it doesn't matter which order you place the factors.
Here are the factor pairs for negative 20. In the first three factor pairs I made the first number negative and the second three factor pairs I made the second number negative. The only factor pair that adds to negative one is four and negative five. So these must be the answer and we could have switched these around. I could have had the 4 over here and the negative 5 on this side too. These are the factor pairs for negative 24. I want to find the factor pair that has a sum of negative 5. That would be 3 and negative 8. So those set is to be my factors. Here are the factor pairs for negative 36. I've only made the first set negative. I know negative 4 and 9 sum to 5 so this should be my answers to the number puzzle. Now if none of these factor pairs added to 5, I would have made this column positive and this column negative. Then I could continue searching.
For the first practice problem, try factoring this expression. We want to put the greatest common factor on the outside and then put what would be left for each term in these boxes.
To factor this expression, we first want to find the greatest common factor. We can find the prime factorization of each of our terms. Now we want to look for the lowest power of each factor that appears in each term. We can see that a factor of 2 appears in each term and in fact there's two of them. So 2 squared is a part of our greatest common factor. The next factor that we multiply into our greatest common factor is 3. There's a power of 3 to the 1 for all of our terms. We can not include x as a factor since it doesn't appear here, but we can include a y. Y to the 1 would be part of our greatest common factor. When I multiply all these factors together I get 12 y. That goes on the outside. When we divide this first term by 12y, we're left with 2x. We can also see that in our factors. If we take out 12y, then we're left with 2x. The second term is a little bit more simple. We will just be left with x times y, and again, we see that in our factors. The last term will be negative 3y squared. It's what we'll get when we multiply the remaining factors together. So this expression has been completely factored. We have one number times another number.
Try factoring this expression. This time write your entire factored form in this box. Your answer should have a greatest common factor on the outside and then terms that are added or subtracted inside the parentheses. Be careful with your exponents and good luck.
Listing out the prime factorizations, we can see that our greatest common factor is 8ab squared. So this is what I list on the outside of a parenthesis to start the factor. We're left with 5ab for the first term. If I factor 8 times ab squared out of the first term, I'm left with 5 times ab. That goes here. Negative 2b squared would be remaining from the second term. And negative 3 would be remaining for this last term. Essentially, we're just dividing this term by 8ab squared. We know ab squared divided by ab squared would equal one, so we're just left with negative 24 divided by 8, which is negative 3. As always, you can check your result by multiplying or distributing this 8ab squared to each of the terms. You should wind up with your original expression.
And finally, what would be the result of factoring this expression? The variables are the hardest part here.
The greatest common factor for 15, 20 and -35 is just 5. That's going to start the product for my GCF. Now I want to lowest factor for f that is common between all the terms. Well, just 1 x is common between all the terms. The lowest power of Y that is common is 2. So y squared would also appear in my GCF. An finally, I look at the lowest power of Z between all of the terms, but I have to be careful. There's no z over here. So z won't be included in my GCF. So a factor of 5x y squared from each of these terms. I need 3x y squared z cubed for the first term. 4z to the fifth of the second term and negative x squared y for the last term. I can do this pretty quickly if just check the multiplication as I go. 5 times 3 is 15. X times x is x squared, and y squared times y squared is y to the 4th. Of course, I also have this z cubed on the end. So this is my first term. When I multiply these two terms together, I get 20xy squared z to the fifth. The variables are just listed are multiplied together. And 5 times 4 is negative 35, and x times x squared is x cubed. The last variable is y cubed, which comes from y squared times y. So yeah, this is definitely my factored form.
Thanks, Chris, for teaching us how to find and factor out the greatest common factor or GCF. For this practice problem, I want you to factor out the GCF and put the GCF in front, and the remaining terms in parentheses following. So we have 27 q squared p cubed, minus 18 q squared p squared, plus 45 q p cubed. Put your answer in this box.
So how do you do? The correct answer is that the GCF is 9qp squared, and the remaining factors are 3qp minus 2q plus 5p. Let's see how we got that answer. First I want to look at the numbers 27, 18, and 45. So I ask myself, what is the largest number that goes evenly into 27, 18, and 45? And with some practice you'll notice that the largest number that goes into those three numbers is 9. When we look at the variables, I need the variable to be in each of the 3 terms and I want to take the lowest exponent of each of the variables. So for the q's, I have q squared, q squared, and q. So the lowest exponent is 1. For the p's I have p cubed, p squared, and p cubed, so the lowest power of p is squared. So the GCF as we said was 9qp squared. What I'm trying to figure out what are the remaining factors, I want to look at the first term, 27 q squared p cubed and I want to ask myself, what do I need to multiply the GCF by to get the first term. First you'll look at the coefficient. I need to multiply 9 by 3 to get 27. I have to multiply q by q to get q squared. And I need to multiply p squared by p to get p cubed. So, that's the first term. For the next term, once again I ask myself. What do I need to multiply the GCF by to get the second term? I need to multiply 9 by 2 to get 18. I need to multiply q by q to get q squared. And what do I need to multiply p squared by to get p squared? Nothing, just 1. So our second term is 2q or negative 2q. And then I put the next sign, which is plus. What do I need to multiply 9qp squared by to get the third term? Well, 9 times 5 is 45. Q times 1 is q. P squared times p gives me p cubed. When factoring, one of the most important steps is to find and factor out the GCF. Let's get back to Chris so he can teach us more about factoring.