 So when we're solving for quadratic equations, which are basically quadratic functions, these things graph. If this is our Cartesian coordinate system, we have the x-axis here and we have y, which is really f of x, this way, right? So for quadratic equations which represent quadratic functions, right, where this is, if we erase this, this thing becomes f of x is equal to ax squared plus bx plus c, that's a quadratic function. And when we're solving for these equations, all we do, we set it equal to zero, right? So whenever we're solving for quadratic functions, we can have either two solutions, two x-intercepts. We could have one x-intercept or we might have no x-intercepts. And the way that works is, now I'm going to draw three different parabolas here, but quadratic equations, quadratic functions, only graph one parabola. So if I draw three different parabolas, that's three different functions, right? But for now, just looking at the solutions, the way it works is, you could have a quadratic function, quadratic equation, quadratic function, crossing the x-axis twice. You could have it only crossing the x-axis once, not crossing it but touching it. So you only have one solution here, right? You could have two x-intercepts, we could have one x-intercept, or we could have a quadratic equation, a parabola, going like this and never crossing the x-axis. And that means we wouldn't have any x-solutions, right? So sometimes when we're solving for these types of functions or for these types of equations, if we already set it equal to zero, if we set the y-axis equal to zero, right, and we're looking for x-intercepts, sometimes when we solve for these, we're going to get no solutions. Sometimes we're going to get one solution, sometimes we're going to get two solutions, okay? And the parabolas don't have to open up, they could open down as well, right? So this one would be a quadratic equation opening down with no solutions because it doesn't cross the x-axis. This would be a quadratic equation opening down with one solution, one x-intercept, or this could be a quadratic equation with two x-intercepts, two solutions. Now when we get into functions, we're going to analyze these a lot more, specifically looking at the function part, not setting it equal to zero. So what we're going to do is analyze these functions and use completing the square manually, how to figure out how to graph these functions, quadratic functions. And they're, you know, it's really straightforward, certain set of rules that you follow. But what we're going to do for now is look at the four different types of factoring techniques, manual factoring techniques where we can factor quadratic equations and solve for the variable for x. And if we can't factor it manually, if it's not that simple to factor manually, what we're going to do is use the quadratic formula, which is x is equal to negative v plus or minus squared of v squared minus 4ac over 2a. And that's going to roll off your tongue by the time you're finished with this series or by the time you're finished grade 12 because you end up using that formula more than any other formula that I've come across in mathematics. That includes, you know, the area of a circle or the area of a triangle or circumference or anything like this, or the Pythagorean theorem. You end up using the quadratic formula way more, or I ended up using it way more than any other formula. So this is where we're going and we're going to acquire basically five additional tools or two sets of tools. One is just straightforward factor and another one is a quadratic formula to solve these types of equations.