Question

nothing

Answer 1

Felix, use the quadratic formula.

Answer 2

Hmm, you just ask Felix to use the quadratic formula. Do you know what the discriminant is?

Answer 3

I respect that. First of all, do you know what the discriminant is?

Answer 4

As in... the meaning of discriminant.

Answer 5

Hmm,

Have you heard of the quadratic formula?

Answer 6

I'm sorry, back.

Answer 7

\[x = \dfrac{-b \pm \sqrt{\color{blue}{b^2 - 4ac}}}{2a}\]

Answer 8

that is the quadratic formula

Answer 9

The above is the quadratic formula. But one part of it is really important: the thing in the root, that is, \(\color{blue}{b^2 - 4ac}\). This quantity is known as the discriminant.

Answer 10

Now, you know how roots work, right? You can't insert a negative in them, or you get jailed.

Answer 11

So similarly, if you insert a negative in the root, you're not gonna get a real solution, would you? That is what we mean by a negative discriminant. If you have a negative \(b^2 - 4ac\) (or negative discriminant), you don't get to see real solutions.

Answer 12

Are you getting it?

Answer 13

Yup, it very much does! Another thing about the discriminant: if it is *zero*, what would the solutions be? Let's see how that goes in the quadratic formula.\[x =\dfrac{ -b \pm \sqrt{\color{blue}{0}}}{2a} = \dfrac{-b}{2a}\]Woot! Only one root.

Answer 14

So, can you create an expression in the form \(ax^2 + bx + c\) where \(b^2 - 4ac\) is negative?

Answer 15

Actually, we can come with random b's, a'c and c's which would make \(b^2 - 4ac\) negative. One way to do it is to make \(4ac\) ridiculously bigger than \(b^2\).

Answer 16

Yuss! You got it!

Answer 17

As the discriminant here is \((4)^2 - 4(2)(10) = 16 - 80 = -64\), this is good.

Answer 18

Yes, the discriminant is -64, so it is negative.

Answer 19

Did you make this expression up on your own? Because it is one of the greatest examples.

Answer 20

OK, great job!

Answer 21

You just use the straightforward quadratic formula here.

Answer 22

Have you learned the concept of imaginary numbers?

Answer 23

Hmm -- you don't really need to understand them too much. But you know what \(i\) is, don't you?

Answer 24

You have\[2x^2+4x + 10 = 0\]so,\[x = \dfrac{-4 \pm \sqrt{16 - 80}}{4} = \dfrac{-4 \pm \sqrt{-64}}{4}\]

Answer 25

Now, what is \(\sqrt{-64}\)?\[\sqrt{-64}\]\[= \sqrt{-1 \times 64} \]\[= \sqrt{-1} \times \sqrt{64}\]\[= i \times 8\]\[= 8i\]

Answer 26

Thanks, but we've got things left to do. :)

Answer 27

\[\dfrac{-4 \pm \sqrt{-64}}{4}\]\[= \dfrac{-4}{4} \pm\dfrac{8i}{4}\]\[= -1 \pm 2i\]

Answer 28

And that, madam, is the end of all the struggles you have had with discriminants.

Answer 29

Thank you for the acknowledgement!

Answer 30

Great, I'll wait for ya!

Answer 31

The equation I came up with that has a negative discriminant is

2x^2 + 4x + 10.

We find the negative discriminant by

x = -4 ± sqrt 16 - 80 / 4 = -4 ± sqrt -64 / 4

Next solve the negative discriminant

sqrt -64

= sqrt -1 * 64

= sqrt -1 * sqrt 64

= i * 8

= 8i

Then continue solving the equation

= -4 ± sqrt -64 / 4

= -4/4 ± 8i/4

= -1 ± 2i

The solution to the equation is -1 ± 2i

Answer 32

Instead of saying
```
We find the negative discriminant by
```

You should say
```
We find the solution of this equation with a negative discriminant by
```

Answer 33

Nothing else! Good job.