BuilderBoi

For the quadratic to have only positive roots,  \(\sqrt{b^2-4ac}\) must be an odd integer. 

This can only happen \(b^2 - 4ac\) is an odd square number. The only ways this can happen is if the discriminant is any 1 of these values: 1, 9, 25, 49, 81, 121, or 169. We can eliminate 225, because then the quadratic would have a root of 0, which isn't positive. 

For the values of the discriminant to happen, t must equal: 14, 26, 36, 44, 50, 54, or 56. From here, you can calculate the mean. 

I can't delete my answer, but this is answer is incorrect. I misread the problem. 🤦‍♂️

Write it as an equation:  \({2x\over5 x-210 } = {10\over11}\)

Cross Multiply:  \(22x = 50x-2100\)

Subtract 50x from both sides: \(-28x = -2100\)

Divide by -28: \(x = 75\)

Because the original ratio was: \(2x \over 5x\), Peter originally had \(\color{brown}\boxed{375}\) VCDs.

Because Peter threw away 210, he threw away \({210 \over 375 }= \color{brown}\boxed{14\over25}\) of his VCDs.

The expected value is the average of all the possibilities:

You can win 0, 1, 2 ,3, 4, or 5 dollars, so the expected is \(\color{brown}\boxed{15\over6}\)