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Discriminant for types of solutions for a quadratic

Video transcript

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 haven't found 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 a little bit that a lot we're going to get different types of solutions depending on what happens under the radical sign over here as you can 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 which and this is what the discriminant really is it's just this is it's just this expression under the radical sign of the quadratic formula if this is greater than zero then we're going to have two real roots then we're going to have I'm gonna have two real two real roots or two real solutions to this equation right here if b squared minus 4ac if b squared minus 4ac is equal to zero then this whole thing is just going to be equal to zero is going to be the plus or minus square root of 0 which is just 0 so it's plus or minus zero 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 one I'll just write the number one one real solution or you could kind of say you have a repeated root here you have you can kind of say you're having it twice or you could say one real one real solution or one 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 where you would get 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 what we have two complex solutions but they will be the conjugates of each of each other so if you have one complex solution for a quadratic equation the other solution will also be a complex solution and will be its complex conjugate so here we would have two we would have two complex complex solutions so numbers that have a real part and an imaginary part and not only are they just complex but they're 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 zero 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 and our first degree term then our constant term and so we can evaluate the discriminant v squared is 5 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 and 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 look we didn't have to do the math we can do we can just say that this is definitely going to be less than zero you can actually figure it out this is equal to this is equal to negative this is equal to negative 13 if I did oh no sorry negative 23 negative 23 which is clearly less than so our discriminant in this situation is less than zero so we are going to have two complex roots here and they're going to be each other's conjugates