When a Quadratic equation does not have real roots

Mon, 17 Jan 2022 13:38:33 +0330

For any equation as $ax^2 + bx + c = 0$ we have $b^2 - 4ac$ or $\Delta$ smaller than zero thus having no real roots if:

$a, c > |b|$
$a = b = c$
$a = b$ and $c > b$ for any positive $b$. You can swap $a$ and $c$.

I have proved these but since it’s trivial, I did not write them here.

A math problem: enriching uranium!

Mon, 17 Jan 2022 13:14:20 +0330

The uranium enriching process is a simple yet hard process. We want to reach 3.5% uranium and we only have got 0.7% uranium. To achieve this goal, we should feed the uranium to a centrifuge. For each 2 units of x% uranium which we feeds to the centrifuge, we gets 1 unit of (x-0.1)% and 1 unit of (x+0.1)% uranium. For example a unit of 0.6% and a unit of 0.8% for 2 units of 0.7%. 0.0% uranium will be discarded. Units are always positive integer and we cannot feed 1 unit of uranium, like to get two $ \frac{1}{2} $ unit.

So to reach 3.5% uranium, 3 questions get raised:

Will we be able to produce 3.5% uranium from $m$ units of 0.7% uranium?
For $m$ units of 0.7% uranium how many units of 3.5% uranium can we produce?
If with some units of 0.7% uranium, we cannot produce at least a unit of 3.5% uranium, what is the richest which we can produce?

You can solve these questions either by your math skills or your coding skills. Good luck :)

Math on Farooq's

When a Quadratic equation does not have real roots

A math problem: enriching uranium!