 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. This is the first lecture in, well, the first video in lecture 17, which we're gonna talk about quadratic equations. We're going to introduce a couple of different methods for solving quadratic equations, and this will continue on into lecture 18 as well. So stay tuned for that. So what do we mean by, of course, a quadratic equation? We've learned a lot about linear equations, quadratic equations are those of the form ax squared plus bx plus c equals zero, or at least any equation that's equivalent to such a thing. So you could, you could move things around the equation to get it to look like this. So some things to notice here that without the ax squared, this is just a linear equation bx plus c equals zero. So to differentiate between the linear equations we studied before in the quadratic equations, we do have to have that the leading coefficient a right here is non-zero. And this is often referred to as the standard form of quadratic equation. We like the right-hand side to equals zero when we work with quadratic equations. We'll make, we'll explain why that isn't just a second. So some examples like we could take the equation two x squared plus x plus eight equals zero, that's a quadratic equation in standard form, three x squared minus five x equals four. It's a quadratic equation not in standard form though, we have to move the four to the other side to get in the standard form. And then this last one as well, x squared plus x equals negative two x plus five. If you put it in standard form, move everything to the left-hand side, you'll get x squared plus three x minus five when you move the negative two x and the five to the other side. So you can always set this equal to zero to put it in standard form. This is the format that we're gonna want to solve this quadratic equation. And the key thing about what makes an equation quadratic is you're gonna have this x squared term with some non-zero coefficient in front of it and there's no other higher powers of x or other things going on here. We're just gonna have the quadratic term, the linear term and the constant term, which admittedly some of these coefficients could be zero, you might have no linear term or a constant term, but to be quadratic you have to have the quadratic term, the x squared and nothing else, nothing else more complicated than that. Now the reason we like the standard form of a quadratic equation is because we can solve quadratic equations using the so-called zero product property. When it comes to real numbers or even complex numbers, the only way that a product of two numbers is gonna equal zero is if one of the factors are equal to zero itself. We know that if you take zero times B, that'll equal zero and if you take A times zero that'll equal zero as well. But the thing is the only way a product of two numbers can produce a zero is if one of the factors are zero itself. And so we can solve quadratic equations using principles of factoring. And so in this lecture, as we try to solve quadratic equations, we're mostly gonna be focusing on this factoring method of solving the equations. And so I wanna use examples to help us practice factoring techniques. So for example, if we look at this one right here, x squared plus two x, whenever you're factoring anything, whether it's a quadratic polynomial or any algebraic expression, the first thing you always wanna look for is the GCD, the greatest common divisor. Look for common factors amongst the terms and then factor them out if you can find them. So if you look at like x squared and a two x, notice they have a common factor of x. So we're gonna factor this x out of the binomial there. And so this is gonna give us x times x plus two, which is equal to zero. And if you have any doubt about whether you have the right factorization or not, you can always multiply it back through, right? Factoring is basically just the opposite of the distributive property, we're just reversing it, okay? And so if you redistribute the x, you'll get x times x, which is an x squared, and then an x times two, which is a two x. This is a correct factorization. Now by the zero product property, or zip or zapper, put whatever vowel sound you want in there, but by zip, we see that the only way this factorization could equal zero is if one of the factors were zero. So it must either be that x is zero or x plus two is zero. Now if x is zero, then we know what x is, but if x plus two is zero, you could subtract two from both sides and get that x equals negative two. And so it's quite typical for a quadratic equation to have two distinct solutions. You get x equals zero and x equals negative two. And we can check the original expression just to make sure here, but if you plug in zero, you're gonna get zero squared plus two times zero. That'll give you zero plus zero is equal to zero, that passes off. If you try negative two, you're gonna get negative two squared plus two times negative two. Now notice negative two squared is actually a positive four because you get negative two times negative two, it's double negative, but then you're gonna get two times negative two, which is negative four, which adds up to be zero as well. So both of these numbers do in fact work. Now one thing I wanna be clear about the zero product property is that the product equals zero only if one of the factors is zero. And so we end up with things like x equals zero and x equals negative two. I'm not saying that x equals zero and x equals negative two. It can't be both numbers simultaneously, but as the numbers are variables, I mean, that's what variable means. It's able to vary. There are different choices we can make for x. And if we choose x equals zero, that is a solution. If we choose x is negative two, that's a solution. And any other choice of x would be a non-solution. But when it's not, I'm not saying that x is simultaneously zero and negative two, that's not a possibility. But these are the only two choices of x that will solve the equation. And we found this using this zero product property that is zipp zippity-zoop.