Integrated question-Quadratics and log

[Leah5467]

Hello,this is the question i don't know. I founnd out the answer,but if f(x)=2 means having a repeated root,then how can it be that 0<x<2? Is it that i misunderstood the concept of discriminant? Thank you!

 

[Otis]

Did you switch from logarithmic form to exponential form? f(x) is an exponent.

k^f(x) = 6x - 3x^2

Substitute 2 for f(x), and solve for k.

?

 

[MarkFL]

I would begin by writing:

[MATH]\log_k(6x-3x^2)=2[/MATH]
This implies:

[MATH]6x-3x^2=k^2[/MATH]
Or:

[MATH]3x^2-6x+k^2=0[/MATH]
Now, as you surmised, we want to look at the discriminant, because if there is only one root, then the discriminant must be zero:

[MATH](-6)^2-4(3)(k^2)=0[/MATH]
[MATH]3-k^2=0[/MATH]
Given that \(0<k\), what must \(k\) be?

 

[Leah5467]

Thank you! Then k must be smaller than the square root of 3?

 

[MarkFL]

Thank you! Then k must be smaller than the square root of 3?

No, by equating the discriminant to zero, we find:

[MATH]k=\pm\sqrt{3}[/MATH]
But, since \(0<k\), we discard the negative root, and we are left with:

[MATH]k=\sqrt{3}[/MATH]

 

[Leah5467]

ohh i see! thank you!

 

[Otis]

... k must be smaller than the square root of 3?

Hi. This is a question that you could answer by using a test value. Pick a value for k that is less than sqrt(3), and see what happens.

EG: Let k = 1

3 - 1 = 0

We see that k = 1 does not work; therefore, k < sqrt(3) cannot be correct.

---

If you don't mind posting the reason why, I'm curious to learn what you were thinking when you originally wrote, "f(x)=2 means a repeated root". (I like to understand how students think.) Cheers

?

 

[Leah5467]

ohh i see. Thank you! I was thinking there was only one root so it means repeated root.

 

[JeffM]

I am not sure about your vocabulary, but if and only if the discriminatant is zero is there something that makes sense to call a repeated root.