AB is diam of circle w/ center O, P on circle, AP extended

[ballaholic8]

Hey guys for this section we covered this is class but as you can see these questions are math contest questions which im having trouble with.

Can someone please explain then to me in detail.

Thanks

 

[tkhunny]

Re: Question

24 is an AWESOME problem!!!

1st thing to notice. If it has TWO Real roots, it must have THREE Real roots. You tell me why.

2nd thing to notice. If it has THREE Real roots, and one of them is a double, it must look like this: $\displaystyle (x-a)^{2}*(2x-b)$. You tell me why.

Expand that expression and equate coefficients on corresponding powers of x. This leads to:

2 = 2

-9 = -b - 4a

12 = 2ab + 2a^2

-k = -ba^2

Solve this system for a and b and you are just about done.

-9 = -b - 4a

12 = 2ab + 2a^2

 

[tkhunny]

Re: Question

Oops! I was supposed to use calculus.

If f(x) = 2x^3 -9x^2 + 12x - k

Then f'(x) = 6x^2 - 18x + 12

Further, f'(x) = 0 at x = 1 and x = 2

Solve for k

f(1) = 0 and f(2) = 0

You tell me why that worked AND produced the same, unique solutions.

Rule of Thumb: Algebra was studied well before the calculus. Don't leave your algebra in your desk drawer.