 We just did one where the discriminant was greater than zero, right, so we had two distinct real roots where the parabola crosses the x-axis in two places where we have two x-intercepts, right, two roots, two factors. Let's do one where the discriminant inside the square root symbol in the quadratic formula is equal to zero, which would mean the parabola, if you're graphing it, the parabola just touches the x-axis, right, just hits it and bounces back up or hits it and bounces back down. I don't think I've ever done one on the ground before, so let's try this one out on the ground. It's fairly simple and you would, again, you wouldn't use the quadratic formula for this one, okay. You would just use simple trinomial factor, but we're going to do it just to show, you know, where the coefficients go in the quadratic formula and it's fairly easy and what's going to happen is the discriminant is going to equal zero, so it's just going to give us one root, one x-intercept, and that one, when the discriminant is equal to zero, it's a double root. It's actually two identical real roots, so what it ends up happening if you're just factoring it, it becomes the same things, the same thing multiplied by itself, okay. Basically it's x squared plus 4x plus 4, right, so what we're going to do is take the coefficients and plug them into a quadratic formula and the quadratic formula, again, is x is equal to negative b plus or minus the square root of b squared minus 4ac over 2a, okay, and you should have that written beside you on a piece of paper that you're always going to be referencing or you're going to memorize it. The best thing to do really is to write it down on a piece of paper because sometimes your mind plays tricks on you and you skip a step and what happens you end up doing it wrong, so super important just to have it form the written beside you. So what we end up having is x is equal to negative 4 plus or minus the square root of 4 squared minus 4 times 1 and that dot there is between the 4 and the 1 is this multiplication symbol, right, so 4 times 1 times 4, right, divided by 2 times 1. Now negative 4 stays as negative 4 plus or minus 4 squared is going to be 16 minus 4 times 1 is just 4 times 4 is going to be 16, so inside the root symbol we're going to have 16 minus 16, right, so it's going to be the square root of 0 and the square root of 0 is just 0, so what we finally end up with is negative 4 divided by 2, so what we got is going to be negative 4 plus or minus square root of 0 divided by 2, so that's just going to be negative 4 divided by 2, right, and negative 4 divided by 2 is just going to be negative 2, so what we got is x is equal to negative 2, right, now this thing was just a straight up factor, if it was equal to 0 your final answer would be negative 2, you're done, right, but this was an expression that said just factor it, right, it didn't say solve it because there is no equal sign there, so what we're going to do is because we want the factors of this expression we're going to grab this negative 2 and bring it over the equal sign, right, to the side where it's on the same side as the x, so we're going to have x and a negative 2 comes where it comes plus 2, so x plus 2 is equal to 0, so we got x plus 2 is equal to 0 and to factor this guy completely from the discriminant, from the quadratic formula, from the discriminant rule or the discriminant property, we know that the discriminant is equal to 0 which means that we have two identical real roots and this is a quadratic, quadratic expression, right, it's quadratic, a quadratic function basically, quadratic equation, but it's a quadratic expression because there is no equal sign, so we know it's going to have to be two things multiplied together, so your answer for this, if you're going to factor this completely, is going to be x plus 2 times x plus 2, okay, so this expression in the top completely factored is going to be x plus 2 all squared, okay, so what we got is x plus 2 all squared and that's the top guy factored completely, okay, again it's fairly simple, all you're doing is just taking your coefficients, plug in the quadratic formula, if your original question was equal to 0, you would leave it right here, you would leave it in this part where x is equal to negative 2, okay, but since this thing is not equal to 0, it's an expression that is basically the question would be asking you to factor it, what you got to do is write it, write out the factors, which is basically bringing whatever numbers are, you know, x equals whatever numbers, if this is a fraction, you got to cross multiply it up to the x and then bring the number over, so whatever's on the other side, you got to bring it to the same side as the x and whatever you have multiplies together to give you the original question, right, the original expression if they ask you to factor it, okay, let's go do another one where the discriminant is actually equal to a negative number and what that tells us is it's, you know, you can't get the root of a negative number, so what that says is you can't factor it, which means in the real number set, in the real number set, because that's where we're functioning right now, in the real number world there are no factors for it, so we can't continue to factor it, so our expression is not going to be factorable, okay, let's do one where the discriminant inside the root symbol is going to be negative and this is one where a hundred percent would have to use the quadratic formula because we don't know how to do it manually, right, that says negative x plus 5, negative x plus 5x minus 8, right, so what we're going to do is plug that into our quadratic formula, so what we got is negative 5 plus or minus 5 squared, which is going to be 25 minus 4 times negative 1 times negative 8, now what we have inside there in the discriminant is negative 4 times negative 1 times negative 8, that's three negatives multiplied together, right, so that's going to be a negative number and 4 times 1 is just going to be 4, 4 times 8 is going to be negative 32, right, at the bottom we got 2 times negative 1, so so what we got right now is negative 5 plus or minus the square root of 25 minus 32 divided by negative 2, now 25 minus 32, that's a negative number, right, so what we have here right now is going to be x is equal to negative 5 plus or minus the square root of negative 7 divided by negative 2 and you can't take the square root of a negative number, right, so right there we know that the above expression is not factorable because we have a negative number inside the root symbol, okay, and if that was equal to zero then there would be no solutions, there would be no x intercepts, there would be no roots, there would be no zeros, no factors, right, so we know from series 3a that anything x squared that's a quadratic equation, a quadratic equation, quadratic functions, graph parabolas, so what that would mean is the parabola doesn't cross the x-axis either even if it's opening up or down