Find the largest integer $k$ such that the equation

5x^2 - kx + 8 - 20x^2 + 45 = 0

has no real solutions.

ChiIIBill Jul 22, 2024

#1**+1 **

First, let's note that if there are no real solutions, then that means the descriminant is less than 0,

Simplifying the quadratic equation a bit, we find that

\(-15x^2-kx+53=0\)

Since the descriminant is b^2-4ac, we have the equation

\(k^2+3180<0\)

This yields \(k^2<-3180\)

Note that this isn't possible. k^2 must be greater or equal to 0, so it cannot be equal to -3180.

I might have done something wrong...not sure, but I think it's impossible

Thanks! :)

NotThatSmart Jul 22, 2024

#2**0 **

No, your answer is correct, NTS.

It's impossible for the discriminant to be < 0

Good job!!!

CPhill
Jul 22, 2024