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- I think we've had some pretty good exposure to the quadratic
- formula, but just in case you haven't memorized it yet, let
- me write it down again.
- So let's say we have a quadratic equation of the
- form, ax squared plus bx, plus c is equal to 0.
- The quadratic formula, which we proved in the last video,
- says that the solutions to this equation are x is equal
- to negative b plus or minus the square root of b squared,
- minus 4ac, all of that over 2a.
- Now, in this video, rather than just giving a bunch of
- examples of substituting in the a's, the b's, and the c's,
- I want to talk a little bit about this part of the
- quadratic formula, this part right there.
- The b squared minus 4ac.
- And we've seen it in a couple of the problems we've done as
- examples, that this kind of determines what our solution
- is going to look like.
- If, for example, b squared minus 4ac is greater than 0,
- we're going to have two solutions, right?
- The square root of some positive number that's
- non-zero, there's going to be a positive and negative
- version of it-- we're always going to have a b over 2a or
- negative b over 2a-- so you're going to have negative b plus
- that positive square root, and a negative b minus that
- positive square root, all over 2a.
- So if the discriminant is greater 0, then that tells us
- that we have two solutions.
- Now I just used a word, and that word is discriminant.
- And all that is referring to is this part of
- the quadratic formula.
- That right there-- let me do it in a different color-- this
- right here is the discriminant of the quadratic equation
- right here.
- And you just have to remember, it's the part that's under the
- radical sign of the quadratic formula.
- And that's why it matters, because if this is greater
- than 0, you're having a positive square root, and
- you'll have the positive and negative version of it, you'll
- have two solutions.
- Now, what happens if b squared minus 4ac is equal to 0?
- If this is equal to 0-- if you take b squared minus 4, times
- a, times c, and that's equal to 0-- that tells us that this
- part of the quadratic formula is going to be 0, and the
- square root of 0 is just 0.
- And then, actually, your only solution is going to be x is
- going to be equal to negative b over 2a.
- Or another way to think about it is you
- only have one solution.
- So if the discriminant is equal to 0, you
- only have one solution.
- And that solution is actually going to be the vertex, or the
- x-coordinate of the vertex, because you're going to have a
- parabola that just touches the x-axis like that, just touches
- there, or just touches like that, just touches at exactly
- one point, when b squared minus 4ac is equal to 0.
- And then the last situation is if b squared minus 4ac
- is less than 0.
- Then over here, you're going to get a negative number under
- the radical.
- And we saw an example of that in the last video.
- If we're dealing with real numbers, we can't take a
- square root of a negative number, so this means that we
- have no real solutions.
- In the future, you're going to see that we will have complex
- solutions, but if we're dealing with real numbers we
- have no real solution.
- Because this makes no sense.
- The square of a negative number, at least it makes no
- sense in the real numbers.
- And then there's more you can think about.
- If we do have a positive discriminant, if b squared
- minus 4ac is positive, we can think about whether the
- solutions are going to be rational or not.
- If this is 2, then we're going to have the square root of 2
- in our answer, it's going to be an irrational answer, or
- our solutions are going to be irrational.
- If b squared minus 4ac is 16, we know that's a perfect
- square, you take the square root of a perfect square,
- we're going to have a rational answer.
- Anyway, with all of that talk, let's do some examples,
- because I think that's what makes all
- of these ideas tangible.
- So let's say I have the equation negative x squared
- plus 3x, minus 6 is equal to 0.
- And all I'm concerned about is I just want to know a little
- bit about what kinds of solutions this has.
- I don't want to necessarily even solve for x.
- So if you're in a situation like that, I can just look at
- the discriminant.
- I can just look at b squared minus 4ac.
- So the discriminant here is what? b squared is 9 minus 4,
- times a-- negative 1-- times c, which is negative 6.
- So what is this equal to?
- This negative and that negative cancel out, but we
- still have that negative out there, so it's 9
- minus 4, times 6.
- This is 9 minus 24, which is less than 0.
- So we're going to have a number smaller than 0 under
- the radical.
- So we have no real solutions.
- That was this scenario right here.
- And so this graph is going to point downwards, because we
- have a negative sign there, so it probably looks like
- something like that.
- If that's the x-axis, the graph is dipping down.
- Its vertex is below the x-axis and it's downward-opening, so
- it never intersects the x-axis.
- We have no real solutions.
- Let's do another one.
- Let's say I have-- I'll do this one in pink-- let's say I
- have the equation, 5x squared is equal to 6x.
- Well, let's put this in the form that we're used to.
- So let's subtract 6x from both sides, and we get 5x squared
- minus 6x is equal to 0.
- And let's calculate the discriminant.
- So, we want to get b squared.
- b squared is negative 6 squared minus 4,
- times a, times c.
- Well, where is the c here?
- There is no c here.
- There's a plus 0 that I'm not writing here.
- There's no c.
- So in this situation, c is equal to 0.
- There is no c in that equation.
- So times 0.
- So that all cancels out.
- Negative 6 squared is positive 36.
- The discriminant is positive.
- You'd have a positive 36 under the radical right there, so
- not only is it positive, it's also a perfect square.
- So this tells me that I'm going to have two solutions.
- So I'm going to have two real solutions.
- And not only are they're going to be real, but I also know
- they're going to be rational, because I have the
- square root of 36.
- The square root of 36 is positive or negative 6.
- I don't end up with an irrational number here, so two
- real solutions that are also rational.
- This is this scenario right there.
- And you could also have irrational in this scenario,
- so it's this [? here ?]
- plus the irrational.
- Let's do a couple more, just to get really warmed.
- Let's say I have 41x squared minus 31x, minus
- 52 is equal to 0.
- Once again, I just want to think about what type of
- solution I might be dealing with.
- So b squared minus 4ac.
- b squared.
- Negative the 31 squared minus 4, times a, times 41, times
- c-- times negative 52.
- So what do I have here?
- This is going to be a positive 31 squared.
- The negative times the negative,
- these are both positive.
- So I'm going to have a positive, right?
- This is the same thing as 31 squared, plus-- this is a
- positive number right here, I mean, we could calculate it,
- but it's 4 times 41, times 52.
- All I care about is my discriminant is positive.
- It is greater than 0, so that means I
- have two real solutions.
- And we could think about whether this is some type of
- perfect square.
- I don't know.
- I'm not going to do it here.
- That would take a little bit of computation.
- So we know they're real, we don't know if they're rational
- or irrational solutions.
- Let's do one more of these.
- Let's say I have x squared minus 8x, plus
- 16 is equal to 0.
- Once again, let's look at the discriminant.
- b squared, that's negative 8 squared minus 4, times a,
- which is 1, times c, which is 16.
- This is equal to 64 minus 64, which is equal to 0.
- So we only have one solution, and by definition it's going
- to be rational.
- I mean, you could actually look at it right here.
- It's x minus 4, times x minus 4 is equal to 0.
- The one solution is x equal to positive 4.
- And when I say by definition of the quadratic formula, you
- look there, if this is a 0, all you're left with is
- negative b over 2a, which is definitely going to be
- rational, assuming you have a, b, and c are, of course,
- rational numbers.
- Anyway, hopefully you found that useful.
- It's a quick way.
- You don't have to go all the way to solving the solution,
- you just want to have to say what types of solutions or how
- many solutions, how many real solutions, or inspect whether
- they're real or rational.
- The discriminant can be kind of a useful shortcut.
- And I also think it makes you kind of appreciate the parts
- of the quadratic formula a little bit better.