graphing q (1 Viewer)

[littleboy]

Find the values of k such that the equation x/(1-x^2) = kx has three distinct roots.

Can anyone show me how to do this both graphically and algebraically plz?/ thanks

 

[sasquatch]

Ok not totally sure if this is correct but i think so...

x / (1-x²) = kx
x = kx(1 - x²)
x = kx - kx³
kx³ - kx + x = 0
x(kx² - k + 1) = 0

x = 0 is one root,

kx² + 1- k = 0

so for three distinct roots, discriminant > 0

(0)² - 4k(1-k) > 0
-4k(1-k) > 0
4k(1-k) < 0

So now using that quadratic solving thingo...

### ____
# # / # # \
----*--------*-------
##/ # # # #\

___0_____1____

so if k > 1, solution holds true, k < 0 solution holds true (by testing both inside and outside of the parabola)

hence k > 1, and k < 0 are the solutions if the original express has 3 distinct roots


If you wanna solve it graphically you first need to sketch y = x / (1-x^2) :

Then draw a line y = kx, where k varies. (itd be best to use your ruler or something, and its easier to do algebraicly)
View attachment 13251

You can clearly see from the graph that for all negative gradients (k < 0) the solution will have three roots. for all values of k, the expression has the root x = 0 (as seen in the algebraic steps), but for values of k > 1, again the expression has three roots.

 

[Riviet]

So now using that quadratic solving thingo...

# \ # # # #/
----*--------*-------
## \_____/

___0_____1____

Just a correction, the graph should be concave down [but your explanation is correct]. ^^

 

[sasquatch]

oh yeah yeah sorry! ill fix that up.. thanks riviet and sorry to the original poster.

 

[Shady01]

yeah Ive seen a simialr question to this but differnet graph lol