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I think we've had some pretty good exposure to the quadratic

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formula, but just in case you haven't memorized it yet, let

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me write it down again.

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So let's say we have a quadratic equation of the

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form, ax squared plus bx, plus c is equal to 0.

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The quadratic formula, which we proved in the last video,

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says that the solutions to this equation are x is equal

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to negative b plus or minus the square root of b squared,

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minus 4ac, all of that over 2a.

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Now, in this video, rather than just giving a bunch of

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examples of substituting in the a's, the b's, and the c's,

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I want to talk a little bit about this part of the

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quadratic formula, this part right there.

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The b squared minus 4ac.

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And we've seen it in a couple of the problems we've done as

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examples, that this kind of determines what our solution

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is going to look like.

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If, for example, b squared minus 4ac is greater than 0,

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we're going to have two solutions, right?

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The square root of some positive number that's

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non-zero, there's going to be a positive and negative

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version of it-- we're always going to have a b over 2a or

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negative b over 2a-- so you're going to have negative b plus

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that positive square root, and a negative b minus that

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positive square root, all over 2a.

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So if the discriminant is greater 0, then that tells us

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that we have two solutions.

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Now I just used a word, and that word is discriminant.

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And all that is referring to is this part of

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the quadratic formula.

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That right there-- let me do it in a different color-- this

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right here is the discriminant of the quadratic equation

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right here.

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And you just have to remember, it's the part that's under the

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radical sign of the quadratic formula.

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And that's why it matters, because if this is greater

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than 0, you're having a positive square root, and

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you'll have the positive and negative version of it, you'll

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have two solutions.

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Now, what happens if b squared minus 4ac is equal to 0?

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If this is equal to 0-- if you take b squared minus 4, times

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a, times c, and that's equal to 0-- that tells us that this

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part of the quadratic formula is going to be 0, and the

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square root of 0 is just 0.

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And then, actually, your only solution is going to be x is

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going to be equal to negative b over 2a.

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Or another way to think about it is you

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only have one solution.

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So if the discriminant is equal to 0, you

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only have one solution.

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And that solution is actually going to be the vertex, or the

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x-coordinate of the vertex, because you're going to have a

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parabola that just touches the x-axis like that, just touches

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there, or just touches like that, just touches at exactly

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one point, when b squared minus 4ac is equal to 0.

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And then the last situation is if b squared minus 4ac

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is less than 0.

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Then over here, you're going to get a negative number under

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the radical.

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And we saw an example of that in the last video.

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If we're dealing with real numbers, we can't take a

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square root of a negative number, so this means that we

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have no real solutions.

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In the future, you're going to see that we will have complex

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solutions, but if we're dealing with real numbers we

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have no real solution.

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Because this makes no sense.

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The square of a negative number, at least it makes no

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sense in the real numbers.

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And then there's more you can think about.

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If we do have a positive discriminant, if b squared

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minus 4ac is positive, we can think about whether the

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solutions are going to be rational or not.

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If this is 2, then we're going to have the square root of 2

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in our answer, it's going to be an irrational answer, or

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our solutions are going to be irrational.

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If b squared minus 4ac is 16, we know that's a perfect

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square, you take the square root of a perfect square,

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we're going to have a rational answer.

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Anyway, with all of that talk, let's do some examples,

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because I think that's what makes all

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of these ideas tangible.

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So let's say I have the equation negative x squared

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plus 3x, minus 6 is equal to 0.

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And all I'm concerned about is I just want to know a little

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bit about what kinds of solutions this has.

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I don't want to necessarily even solve for x.

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So if you're in a situation like that, I can just look at

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the discriminant.

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I can just look at b squared minus 4ac.

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So the discriminant here is what? b squared is 9 minus 4,

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times a-- negative 1-- times c, which is negative 6.

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So what is this equal to?

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This negative and that negative cancel out, but we

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still have that negative out there, so it's 9

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minus 4, times 6.

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This is 9 minus 24, which is less than 0.

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So we're going to have a number smaller than 0 under

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the radical.

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So we have no real solutions.

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That was this scenario right here.

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And so this graph is going to point downwards, because we

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have a negative sign there, so it probably looks like

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something like that.

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If that's the x-axis, the graph is dipping down.

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Its vertex is below the x-axis and it's downward-opening, so

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it never intersects the x-axis.

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We have no real solutions.

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Let's do another one.

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Let's say I have-- I'll do this one in pink-- let's say I

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have the equation, 5x squared is equal to 6x.

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Well, let's put this in the form that we're used to.

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So let's subtract 6x from both sides, and we get 5x squared

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minus 6x is equal to 0.

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And let's calculate the discriminant.

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So, we want to get b squared.

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b squared is negative 6 squared minus 4,

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times a, times c.

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Well, where is the c here?

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There is no c here.

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There's a plus 0 that I'm not writing here.

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There's no c.

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So in this situation, c is equal to 0.

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There is no c in that equation.

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So times 0.

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So that all cancels out.

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Negative 6 squared is positive 36.

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The discriminant is positive.

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You'd have a positive 36 under the radical right there, so

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not only is it positive, it's also a perfect square.

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So this tells me that I'm going to have two solutions.

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So I'm going to have two real solutions.

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And not only are they're going to be real, but I also know

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they're going to be rational, because I have the

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square root of 36.

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The square root of 36 is positive or negative 6.

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I don't end up with an irrational number here, so two

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real solutions that are also rational.

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This is this scenario right there.

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And you could also have irrational in this scenario,

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so it's this [? here ?]

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plus the irrational.

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Let's do a couple more, just to get really warmed.

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Let's say I have 41x squared minus 31x, minus

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52 is equal to 0.

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Once again, I just want to think about what type of

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solution I might be dealing with.

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So b squared minus 4ac.

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b squared.

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Negative the 31 squared minus 4, times a, times 41, times

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c-- times negative 52.

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So what do I have here?

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This is going to be a positive 31 squared.

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The negative times the negative,

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these are both positive.

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So I'm going to have a positive, right?

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This is the same thing as 31 squared, plus-- this is a

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positive number right here, I mean, we could calculate it,

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but it's 4 times 41, times 52.

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All I care about is my discriminant is positive.

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It is greater than 0, so that means I

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have two real solutions.

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And we could think about whether this is some type of

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perfect square.

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I don't know.

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I'm not going to do it here.

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That would take a little bit of computation.

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So we know they're real, we don't know if they're rational

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or irrational solutions.

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Let's do one more of these.

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Let's say I have x squared minus 8x, plus

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16 is equal to 0.

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Once again, let's look at the discriminant.

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b squared, that's negative 8 squared minus 4, times a,

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which is 1, times c, which is 16.

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This is equal to 64 minus 64, which is equal to 0.

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So we only have one solution, and by definition it's going

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to be rational.

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I mean, you could actually look at it right here.

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It's x minus 4, times x minus 4 is equal to 0.

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The one solution is x equal to positive 4.

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And when I say by definition of the quadratic formula, you

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look there, if this is a 0, all you're left with is

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negative b over 2a, which is definitely going to be

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rational, assuming you have a, b, and c are, of course,

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rational numbers.

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Anyway, hopefully you found that useful.

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It's a quick way.

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You don't have to go all the way to solving the solution,

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you just want to have to say what types of solutions or how

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many solutions, how many real solutions, or inspect whether

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they're real or rational.

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The discriminant can be kind of a useful shortcut.

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And I also think it makes you kind of appreciate the parts

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of the quadratic formula a little bit better.

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