# Thread: Quadratic Inequalites

1. ## Quadratic Inequalites

Hi. First time posting, and with a big A level exam coming up tommorow - I hope to make it a good first post

In every single paper so far, there seems to be a Quadratic Equation (I think a Quadratci inequality question?). Heres an example of one:

"The quadratic equation (k+1)x^2 + 12x + (k-4) = 0 has real roots.

(a) Show that k^2 - 3k -40 < 0... See more
(b)Hence find the possible values of K."

If someone could walk be through both questions it would be greatly appreciated. Could meant the difference between a grade or 2!

2. Originally Posted by spongebob92
Hi. First time posting, and with a big A level exam coming up tommorow - I hope to make it a good first post

In every single paper so far, there seems to be a Quadratic Equation (I think a Quadratci inequality question?). Heres an example of one:

"The quadratic equation (k+1)x^2 + 12x + (k-4) = 0 has real roots.

(a) Show that k^2 - 3k -40 < 0... See more
(b)Hence find the possible values of K."

If someone could walk be through both questions it would be greatly appreciated. Could meant the difference between a grade or 2!
I assumed that k is a constant

(a) Use the discriminant: $\Delta = b^2-4ac$. Since you want to show there are two real roots it must be greater than 0.

I'm not too sure about why you'd show it's greater than 0, hence the above is the best bet. I would assume that since the question says there are real roots it's fine to assume part a but meh,perhaps someone else knows more?

(b) Solve the equation given in part a for k using your favourite method

3. So, using $b^2 - 4ac$ you would get $144-4(k+1)(k-4)$.

I have no idea how you get any further?

4. for the equation (k+1)x^2 + 12x + (k-4) = 0 to have real roots, you need to have 12^2 - 4(k-4)(k+1) positive.
thus you need to have 4k^2-12k-140>=0

5. papex, how do you get the 12K in the above?

6. I am getting:

12^2 - 4(k-4)(k+1) =
144 - 4(k^2-3k-4) =
144 - 4k^2+12k+16 =
k^2-3k-40

7. it comes from the calculation of the discriminant (b^2-4ac) which is equal to:
12^2-4(k-4)(k+1)=144-4k^2+12k+16=160-4k^2+12k (there was a mistake in my previous post. sorry)
for the equation to have real roots you need to have the above positive which is equivalent to have 40-k^2+3k positive

8. Where about do you get the 12K in the above example?

(Thank for the continued help)

9. Multiplying out of brackets works as follows:

(k-4)(k+1) = k*k-4*k+1*k-4*1 = k^2-3k-4

-4(k-4)(k+1) = -4(k^2-3k-4) = -4k^2+12k+16