soft quadratic equations by factory. Graphing is one way you can solve them, but it's not the only way. Sometimes you can also solve by factory. So let's factor these quadratic equations and remember to factor. Um, we want them all to equal zero. So we have one here, five X squared, minus 13 X plus six. Now, this is an example where I have a coefficient that's greater than one for my a. So we're gonna get this one's gonna take a couple more steps than a one That doesn't happen. So let's go ahead and work it now The way I teach my students to do it is we take that first coefficient and we bring it over and we multiply it by our sequel coefficient. And by doing so, I get X squared minus 13 X top plus 30 equals zero. So this is how we're going to start factoring and we're gonna look for two factors of 30. Now, when I add them together will give me a negative 13. Well, since they're both gonna add up and give me a negative number, that tells me that one's gonna have to be negative. And since my multiplication is positive. That just tells me both of them have to bay. So if I listed the factors of 30 that's one way you could start. So two factors of 30 that we think of real quick. Let's do six and five. Well, six times five does give me 30 six plus five gives me positive 11. Well, even if I made both of those negative, I'm still going to get the 11. So let's try another factor. Well, let's do three and 10. Well, three times 10 is 33 plus tennis 13. Well, it's not exactly what I want, because I won't Negative 13. So that's an easy fix. We're just gonna let's make them both negative. Negative. Three times negative tennis. Still positive. 30. And when I add them together, I get negative. 13. So when my to parentheses, I'm gonna have X minus three and then x minus 10. Now, remember how we multiply this five? We gotta bring it back since we multiplied it by moved, moved it over by multiplying it. We gotta do the opposite to move it back. So that means we're going to divide each of these by five and we're going to simplify. Well, negative 3/5 will not simplify. So if it won't simplify, my denominator is gonna move in front of my ex. So I'll have five X minus three. Negative. 10/5 will simplify. So in my second parentheses, I'll have X minus two. So if it doesn't simplify to a whole number, you're gonna move the denominator in front of that X. So both of those will equal zero. So now that we know that five X minus three and X minus two or the two factors we can finish solving for the equation So all you're gonna do is now that you know each factor, you're gonna make each factor equal zero. So let's start with five X minus three equals zero, and we're going to solve for X, so we'll add three to both sides. So five X equal three. When we're gonna divide by five, so X is going to equal three fists. That's one solution for X. For the next one, we're going to say that X minus two equals zero, and we're gonna add to so X will equal to Let's look at the quadratic equation X squared plus five x minus 24 equals zero. So for this one, we're going to kind of go the same way, except this time are a equals one. So we're not gonna have to do the self's of moving that coefficient over. Instead, we can start by finding are factors. So we wanna find two factors of R C, which is negative. 24. They're gonna add together and give me five. If you're not sure, the best way is to start by listing the factors. So we know negative 24 1 would give me negative 24 but they're going to give me a negative 23 so I would continue going to save some time. I'm gonna kinda help out. So we would get to eight and negative three, eight and negative three. Giving negative 24. And when I add because the eight is positive and it's the larger number is going to give me positive five. So I would have X plus eight and X minus three. So now that I've family, two sets of factors, we're gonna make each of them equal zero. So we have X plus eight equals zero, and we would solve for X, so X would equal negative eight and then we'd have X minus three equals zero and we would add three to both sides and make it equal three. So are two solutions would be negative. Eight. Positive three.