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Using the quadratic formula: number of solutions

Video transcript

determine the number of solutions to the quadratic equation x squared plus 14x plus 49 is equal to zero there's a bunch of ways we could do it we could factor it and just figure out the values of X that satisfied and just count them that'll be the number of solutions we could just apply the quadratic formula but what I want to do here is actually explore the quadratic formula and think about how we can determine the number of solutions without even maybe necessarily finding them explicitly so the quadratic formula tells us that if we have an equation of the form ax squared plus BX plus C is equal to 0 that the solutions are going to be or the solution if it exists is going to be negative B plus or minus the square root of b squared minus 4ac all of that over 2a now the reason why this is two solutions or the reason why this can be two solutions is that we have a plus or minus here if this b squared minus 4ac is a positive number so let's think about this a little bit if b squared minus 4ac is greater than zero what's going to happen well then it's it's a positive number it's going to have a square root and then when you add it to negative B over 2a or when you add it to negative B you're going to get one value for the numerator and when you subtract it from negative B you're going to get another value in the numerator so this is going to lead to two solutions two solutions now what happens if if b squared minus 4ac is equal to zero if this expression under the radical is equal to zero or just going to have the square root of zero so it's going to be negative B plus or minus zero and doesn't matter what you add or subtract zero you're going to get the same value so in that situation the actual solution of the equation is going to be negative B over 2a there's not going to be this plus or minus it's not going to be relevant so you're only going to have one solution so if b squared minus 4ac is equal to zero you only have one solution and then what happens if b squared minus 4ac is less than zero well if b squared minus 4ac is less than zero this is going to be a negative number right here and you're going to take the square root of a negative number and we know if we're dealing with real numbers you can't take the square root there is no real number squared that becomes a negative number so in this situation there is no solutions or no real when I say real I literally mean like a real number no real solution no real solution so let's think about it in the context of this equation right here and just in case you're curious if whether this this this expression right here b squared minus 4ac has a name it does it's called the discriminant this criminal discriminant this is the discriminant that's that part of the quadratic equation it determines the number of solutions we have so if we want to figure out the number of solutions this equation we we don't have to go through the whole quadratic equation although it's not that much work we just have to evaluate b squared minus 4ac so what is b squared minus 4ac so b is right here it's 14 so it's 14 squared minus 4 times a which is 1 times C which is 49 that's C right there times 49 well what's 14 times 14 14 you do it over here 14 times B 14 times 14 4 times 4 is 16 4 times 1 is 4 plus 1 is 56 put a 0 1 times 14 is 14 it is 6 9 1 it's 196 so this right here is 196 and that's what's we could ignore the 1 what's 4 times 49 so 49 times 4 4 times 9 is 6 sorry it's 36 not 46 it's 36 4 times 4 is 16 plus 3 is 190 or is 19 so you get 196 so this right here is 196 so B squared minus 4 AC is 196 minus 196 so 196 minus 196 is equal to 0 so we're dealing with a situation where the discriminant is equal to 0 we only have one solution and if you want you could try to find that one solution this whole part is just going to be the square root of zero it's just going to be zero so the solution is going to be negative B over 2a and negative B is we could just solve it negative B is negative 14 over 2 times a a is just 1 over 2 so it's equal to negative 7 that's the only solution this equation but if you just wanted to know how many solutions you just have to find out that B squared minus 4ac is 0 so it's only going to have one solution and there's other ways you could have actually factored this pretty easily into X plus 7 times X plus 7 and gotten the same result