circles

[tawdry pendant]

Where did 6 - root 36-4k come from?

[crimson imp]

U have a quartic but all 4 soltutions need to be real. U can notice that since theres no cubic term if we subsitute =t^2 for u we can solve for u and then squareroot. For there to be 4 solutions u must not equal to 0 so we look for conidtions where this is true '

We first ensure the discriminant of u^2 - 6u + k is greater than 0 and then we ensure that 6 +/- sqrt(36 - 4k) is postive

6 - root 36-4k comes from the discriminant / quadratic formula

[tawdry pendant]

Where did u come from?

[lofty vector]

thats not nice

hes tryna help you

[crimson imp]

The discord

[tawdry pendant]

Bro i mean the letter u😭

[crimson imp]

ohh thats just choosing a variable

ill show you one sec

[tawdry pendant]

Thanks

[lofty vector]

oh

[tawdry pendant]

Thank you for the working but I’ll just firm the marks

I don’t get it😭

[crimson imp]

Just look for conditions that need to be true to have 4 roots that are real
If you square root smth less than 0 you will have a complex solution so just find and make sure that the inside of square roots are real

[bitter pilot]

what paper is this?

[tawdry pendant]

2024

[bitter pilot]

where do you find it

[tawdry pendant]

Bought it off the exam board

[lofty vector]

😱

[bitter pilot]

how much did you buy it for

[tawdry pendant]

Its the mock set 4 papers😭

[steep swan]

@tawdry pendant you could have used Discriminant as they did

[tawdry pendant]

I only got k>-9 when i used the discriminant

[steep swan]

Also I think 0 in this case there is only k

[steep swan]

Lmao

[tawdry pendant]

Wdym

[polar tundra]

what paper is this from

[polar tundra]

OH

AS?

[tawdry pendant]

@crimson imp can we go back to this? Im trying to understand this now

So you subbed in the parametric eqns into cartesian eqn to get a quartic

Im not understanding the unique values of u bit

[crimson imp]

Yes

[crimson imp]

Youre are looking for conditions for which you will have 4 unique values of t

No repeats and all real

[tawdry pendant]

Right

So just doing the discriminant gives you two?

[tawdry pendant]

So

$$t=\sqrt{\frac{6+-\sqrt{36-4k}}{2}$$

[odd cradle]

penaldo3142
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[crimson imp]

Yh so basically check the inside of both of the square roots and what are the requirements rhat it is not complex

And gives 2 values

[crimson imp]

Also technically its +/- for both sqrts

[tawdry pendant]

K>-9 is easy to spot

[crimson imp]

$$t=-sqrt{\frac{6+-\sqrt{36-4k}}{2}$$

[tawdry pendant]

But less than 0?

$$t=\pm\sqrt{\frac{6\pm\sqrt{36-4k}}{2}$$

[odd cradle]

penaldo3142
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[crimson imp]

Ok but then consider the inside off the outer sqrt that cant be negative either

So what condition has to be set there

[tawdry pendant]

Doesn’t k>-9 still apply for that since it’s just be 6/2

K = 9 can’t it why does it have to be less than 0

You’ve disappeared😭

[crimson imp]

Think about what happens if 6-sqrt(36-4k) is negative what happens?

[tawdry pendant]

Cant square root a -ve

[crimson imp]

I went to math discussion

[crimson imp]

Exactly so now what does this mean 6-sqrt(36-4k) has to be

[tawdry pendant]

Positive

Ohhhhh

[crimson imp]

Now keep with that idea and find another condition for k

[tawdry pendant]

I see

That gives me k<0

[tawdry pendant]

Btw what made you use the quadratic formula?

[crimson imp]

t^4-6t^2+k=0 is what i like to call a fake quartic

Its a quartic but notice that t^4 is (t^2)^2

So you can assign t^2 to some generic variable like u

[tawdry pendant]

Igy sort of like a hidden quadratic

[crimson imp]

Yh

And then you can use quadratic formula and then solve using +/- sqrt

[tawdry pendant]

Then I just use the discriminant for the other range?

[crimson imp]

Pretty much

[tawdry pendant]

Alright thank you sm

[crimson imp]

Cause the first sqrt also cant be negative

[crimson imp]

Nws

[tawdry pendant]

Wait one more thing😭

$$\sqrt{36-4k}\textgreater0$$
$$36\textgreater4k$$
$$k\textless9$$

[odd cradle]

penaldo3142

[tawdry pendant]

Not minus 9😭

[crimson imp]

It should be sqrt(36+4k)

[tawdry pendant]

Why plus 4k?

[crimson imp]

t^4-6t^2-k=0

[crimson imp]

I copied it down wrong earlier here 💀

[tawdry pendant]

Oh are we not considering the inner sqrt here?

Ohhh lol

Aight thank you