Solving Quadratic Equations, Part 3 5 - 1c

This is going to be another video about solving a quadratic equation by factoring, and I just went to do a couple more examples where you might run into some things that weren't covered in the first few videos. So here we go... What I've got here is 3x squared minus 3x equals x squared plus 2. And, it doesn't look like a quadratic equation at first, but if you think about it, I've got an x squared and an x squared. I can combine these terms and my highest exponent, my largest explicitly is gonna be a 2. So, I can put this into standard form, making it into a nice quadratic equation. What I want to do is get everything on one side, all the terms on one side. So I'll subtract x squared from both sides... and you know you can subtract more than one thing at a time, so let's also subtract from the 2 from both sides. Let's see what I get. 3x squared minus x squared will be 2x squared. 3x... I'm not subtracting anything from it, so I've still got that negative 3x, and I'm subtracting negative 2 from this side, so I get a negative 2. On the right side I'm subtracting x squared minus x squared, so they cancel, and 2 minus 2 cancels. So I've got my zero over on the right and now I've got a quadratic equation in standard form. And I want to factor this left side. Looking at it, realize what I've got is a first coefficient which is greater than 1, so I'd probably want to use the AC method. So I'm going to think of the number 2, the coefficient 2, as an A, the negative 3 is going to be the B, negative 2 will be the C. And I want to multiply A times C. So 2 times negative 2 is negative 4, and for the B I've got a negative 3. So I'm looking for two factors of negative 4 that I can multiply together and get a negative 4,and add together and get a negative 3. So I know that one is positive and one is negative. So let's see what factor I have for 4. Let's start with 1 and 4, and this is nice because 1from 4 is 3, so I know their difference is 3. I've just got to figure out which one should be positive and which one should be negative. Since I want them to add up to negative 3, the 4 should be the negative number. So I'm going to have a 1 and a negative 4. And now I'll replace this negative 3 with the 1 and negative 4. So I'm going to get 2x squared... I'm gonna start with my negative number... minus 4x plus 1x... because this negative 4 and x add up to negative 3x... minus 2. That equals zero. And now I want to factor this by grouping. So I'm gonna group the first two terms and the second two terms. I can factor a 2x out of the first binomial. So I'm going to get 2x times x minus 2. And the second one... really the only thing I can factor out is a 1. So let's put a 1 there. Okay, I've got 2x times x minus 2 plus 1 times x minus 2, so since I've got an x minus 2 and an x minus 2, I'm going to make x minus 2 one of my factors. And my other factor will be 2x plus 1. And that equals zero. Now I'm going to take each of the factors and set it equal to zero. So I get x minus 2 equals zero and 2x plus 1 equals zero, and solving the first one for x, I just add 2 to both sides, and I get x equals 2. Solving the second one, I'm gonna subtract 1 from both sides. I'll get 2x equals negative 1, and then I divide both sides by 2 and I'm going to get x equals negative 1/2. So my answers should be x equals 2 and negative 1/2. Now we've got a fraction here... that shouldn't stop you from checking your answer. So (Let's get something else to write on) and we'll check the answer. Okay, let's see. The original problem was 3x squared minus 3x equals x squared plus 2, and I want to see if the answers are x equals 2 and negative 1/2. So I'm gonna plug the 2 in first. And I have 3 times 2 squared minus 3 times 2, and the question is does that equal 2 squared plus 2. So let's see what happens. 2 squared is 4, 3 times 4 is 12, 3 times 2 is 6. So I've got 12 minus 6. 2 squared is 4 plus 2. 12 minus 6 is 6, 4 plus 2 is 6, so this answer checks, the x equals 2. Now we'll do the fraction part. A lot of students don't like fractions, and because they don't like them, they ignore them and they avoid them, and then when you avoid them, they become harder. So, let's just do it. Okay, 3 times negative 1/2 squared minus 3 times negative 1/2. We want to see if that equals negative 1/2 squared plus 2. So I'm squaring negative 1/2. When you square a negative number, it's positive, so I don't have to worry about this negative sign here. I'm squaring a fraction, which means I'm distributing this 2 to the numerator and the denominator. So I'm going to get 3 times... 1 squared is 1, and 2 squared is 4, minus... I've got a minus 3 times minus 1/2, so that's plus... 3 times 1/2... I just multiply the numerators... 3 times 1 is 3, over 2. Negative 1/2 squared... Well we did that before, that's 1/4 plus 2. Okay, let's see what we can do. It looks like my common denominator.... Actually, let's just multiply this 3 times 1/4. That's going to be 3/4 plus 3/2 equals 1/4 plus 2, and I'm gonna write that as 2 over 1, so I can think of everything as a fraction. My common denominator is 4, So I'm going to multiply everything by 4. These two 4's will cancel, I'll get 3. I can factor out a 2 here and I'll get 2 times 3 is 6. These 4's will cancel and I'll get a 1, 4 times 2 is 8. (I have to keep writing the question mark.) 3 plus 6 is 9, 1 plus 8 is 9, so both of my answers check. Okay, I think that's about all I have time for, so good luck with all this, do plenty of practice. I'll see you next time.