that the discriminate and the discriminate is part of the quadratic formula. Now the quadratic formula says that X square my equals X equals negative B plus or minus B squared minus four a. C over two A. Well, the discriminate is was inside that square right side. The discriminate equals B squared minus for a C. And the reason we use this is this actually helps us know what, how many solutions and what kind of solutions we're gonna have to the quadratic equations. So let's look at what I may. Well, if my discriminate is greater than zero and a perfect square, we're gonna have to rational riel roots If it is greater than zero. But non perfect, we're gonna have to irrational real roots. So if it's greater than zero, then you wanna look, is it gonna be a perfect square or not? And the decide if it's a rational answer or irrational answer? If it equals zero exactly, then means we're only gonna have one room, one solution, And if it's less than zero, we'll have two complex numbers, and this is the reason why it would be a square root of a negative number. So it could be a a perfect square. That's negative, But it's still gonna be a complex number. So this is gonna be an imaginary number. So let's look at a few. And all we're going to be determining is the discriminate and how many solutions. So we have non X squared, modest 12 eggs plus four. Well, the first thing I want to know is what is my A What is my B and what is my c? Well, my a s nine, my Biest Negative 12. And my C is four. The discriminate, remember is B squared minus for a C. So we'll use all three of these numbers, so b squared is gonna be negative. 12 squared minus four times nine times four and that's going to give me this is gonna be negative. 12 squared is 144 negative. Four times nine times four is 144. So 144 minus 144 is zero. So this means that my discriminate equals zero. Well, according to my table, if it equals zero, it is a one route answer. So this one would have won route or one solution. Let's look at this 12 X squared minus 16. Eggs plus 33 are A is to RBS negative 16 and R. C is 33. Our discriminate is B squared minus for a C. So we're gonna have negative 16 squared minus four times, two times 33. So negative 16 squared is 256 Modest. Four times. Two times 33 is 264 which equals negative eight negative. Eight is less than zero. So if it's less than zero is gonna be a complex number. And that reason why is if I was using the quadratic formula, that would give me squaring of negative eight, which we know is gonna be an imaginary number. So this would be too complex numbers. Now, let's look at negative five X squared plus eight X minus one again, we're gonna start by defining or a B and C. A. S negative. Five b is eight. C is native one. The discriminate says B squared minus for a C. So we're gonna have eight squared minus four times negative five times negative one. So that's 64 modest 20 which is positive 44. Now 44 s greater than zero. When the discriminate is greater than zero, you have one more thing to look at. Is it a perfect square or not? 44 is not a perfect square. So according to my table, if it's not a perfect square, it's going to have to Irrational riel roots and the other word for roots are solutions. We call them roots or zeros. However, you will have the final. And finally, let's look at one more negative seven X plus 15 X squared minus four. Now this one is written kind of out of order, so let's put it in order. First of all, so let's rewrite it with our 15 X squared first minus seven X minus four equals zero. Now you really don't have to rewrite it if you know which terms go with which. But make sure that you're writing the right terms. So R. A S 15 R B is negative. Seven and R C is negative. Four 15 is a because it goes with the X Square. The discriminate says that it's B squared minus for a C, so we're gonna have a negative seven squared minus four times, 15 times negative. Four. So we're going to have 49 minus 112. And that's gonna be a negative 63. Well, negative. 63 is less than zero. So this is has to complex numbers as the solution.