Nature of roots
Discriminant for types of solutions for a quadratic
Use the discriminant to state the number and type of solutions for the equation negative 3x squared plus 5x minus 4 is equal to 0. And so just as a reminder, you're probably wondering what is the discriminant. And we can just review it by looking at the quadratic formula. So if I have a quadratic equation in standard form, ax squared plus bx plus c is equal to 0, we know that the quadratic formula, which is really just derived from completing the square right over here, tells us that the roots of this, or the solutions of this quadratic equation are going to be x is equal to negative b plus or minus the square root of b squared minus 4ac, all of that over 2a. Now, you might know from experience applying this little bit, we're going to get different types of solutions depending on what happens under the radical sign over here. As you could imagine, if what's under the radical sign over here is positive, then we're going to get an actual real number as its principal square root. And when we take the positive and negative version of it, we're going to get two real solutions. So if b squared minus 4ac-- and this is what the discriminant really is, it's just this expression under the radical sign of the quadratic formula. If this is greater than 0, then we're going to have two real roots or two real solutions to this equation right here. If b squared minus 4ac is equal to 0, then this whole thing is just going to be equal to 0. It's going to be the plus or minus square root of 0, which is just 0. So it's plus or minus 0. Well, when you add or subtract 0, that doesn't change the solution. So the only solution is going to be negative b over 2a. So you're only going to have one real solution. So this is going to be 1-- I'll just write the number 1-- 1 real solution. Or you could say you have a repeated root here. You could say you're having it twice. Or you could say one real solution, or one real root. Now, if b squared minus 4ac were negative, you might already imagine what will happen. If this expression right over here is negative, we're taking the square root of a negative number. So we would then get an imaginary number right over here. And so we would add or subtract the same imaginary number. So we'll have two complex solutions. And not only will we have two complex solutions, but they will be the conjugates of each other. So if you have one complex solution for a quadratic equation, the other solution will also be a complex solution. And it will be its complex conjugate. So here we would have two complex solutions. So numbers that have a real part and an imaginary part. And not only are they just complex, but they are the conjugates of each other. The imaginary parts have different signs. So let's look at b squared minus 4ac over here. This is our a, this is our b, and this is our c. Let me label them. a, b, c. And I can do that, because we've written it in standard form. Everything is on one side. Or in particular, the left hand side. We have a 0 on the right hand side. We've written it in descending, I guess, power form, or the descending degree. Or we have our second degree term first, then our first degree term, then our constant term. And so we can evaluate the discriminant. b is 5. So b squared is 5 squared, minus 4 times a, which is negative 3, times c, which is negative 4-- I have to be careful. c is this whole thing. c is negative 4. And so I don't know if I said 4 earlier, but c is negative 4. We have to make sure we take the sign into consideration. So times c, which is negative 4 over here. And so this is 25. And then negative 3 times negative 4, that is positive 12. And then 4 times 12 is 48. But we have a negative out here. So 25 minus 48. And 25 minus 48, we don't even have to do the math. We can just say that this is definitely going to be less than 0. You can actually figure it out. This is equal to negative 13, if I did-- oh no, sorry, negative 23, which is clearly less than 0. So our discriminant in this situation is less than 0. So we are going to have two complex roots here, and they're going to be each other's conjugates.