I need help with an inequality question

1. Oct 2, 2004

Gughanath

what are the range of values of k that gives the equation (k+1)x^2+4kx+9=0 ...I work it out ...please help

2. Oct 2, 2004

arildno

Welcome to PF!
I think you have omitted something from your text; what was that?

3. Oct 2, 2004

Gughanath

ups sorry...it should say at the end...that gives the equation no real roots

4. Oct 2, 2004

arildno

And what do you think that means?

5. Oct 2, 2004

shmoe

Hi, you can use the quadratic formula to find the roots of your equation. Forcing no real roots is equivalent to forcing the discriminant (the part under the root sign) to be negative. This will give you conditions on k that you're looking for.

ps. you have to make a restriction on k to guarantee that your equation is a quadratic and not linear. what is this restriction?

6. Oct 2, 2004

Gughanath

i cant use the quadrativ formula for this euqation..when i work out the bracket i get

7. Oct 2, 2004

arildno

Let a=k+1, b=4k, c=9.
ax^2+bx+c=0
Can you solve that one?

8. Oct 2, 2004

Gughanath

i know that the discrimiant has to be smaller than 0....

9. Oct 2, 2004

arildno

And what is the discriminant, expressed with a,b and c?

10. Oct 2, 2004

Gughanath

no i am still confused

11. Oct 2, 2004

arildno

12. Oct 2, 2004

Gughanath

could you please just show me how i work out this questions?

13. Oct 2, 2004

Gughanath

because then i will undertand

14. Oct 2, 2004

arildno

Given the equation:
ax^2+bx+c=0
What is the discriminant?

15. Oct 2, 2004

Gughanath

16k^2-(4[k+1]*9)<0 that becomes 16k^2-36k-36<0

16. Oct 2, 2004

shmoe

Good, now can you find the values of k that satisfy this new inequality?

17. Oct 2, 2004

arildno

Very good!
Here's a hint:
In order to find the range of k-values your after,
1. find the zeros in your discriminant.
That is, solve the equation for k:
$$16k^{2}-36k-36=0$$
2. You weren't interested in the k-values for which the discriminant was zero, but the k-values for which the discriminant is less than zero.
But you should figure out for yourself that those values must lie between the two values found in 1.

18. Oct 2, 2004

Gughanath

i see that i have to factoris ethe equation in the disriminant now..but i cant find the right numbers

19. Oct 2, 2004

arildno

Quite true!
Look at my previous post for hints.

20. Oct 2, 2004

Gughanath

please...i have no idea how to continue...

21. Oct 2, 2004

arildno

Well, what values of k solves:
$$16k^{2}-36k-36=0$$ ?

(Note: You were asked to find the values of k so that the discriminant is less than zero, not zero, but finding the zeroes is a good start)

22. Oct 2, 2004

Gughanath

23. Oct 2, 2004

arildno

Now, having found the k-values yielding zero discriminant, you should be able to write the discrimanant as:
$$16k^{2}-36k-36=16(k-3)(k+\frac{3}{4})$$
What must then the k-interval be which yields negative discriminant?

24. Oct 2, 2004

Gughanath

how did u factorise that?

25. Oct 2, 2004

arildno

It is the roots of equation gained by setting the discriminant equal to 0 (that is, -3/4 and 3 are the roots)