When will the goal post moving stop?

The fact that college professors are actively trying to fuck me over in every test really discourages me.
What's the point on this "FAIL FAIL FAIL" atmosphere in every STEM course? Fuck me I just want to study but this drains every motivation out of my body.

I'm given the following quadrating equation:
$ax^2+bx+c = 0$
and im asked to find a c such that there is only one solution to the equation.

I know that the solutions to a quadratic equation can be found with this formula:
$x = \frac{(-b \pm \sqrt{ b^2 - 4ac } )}{2}$
which can have 2 solutions $x_-$ and $x_+$

so for there to be just one solution $x_-$ must equal $x_+$
so i took this and derived the following:
$<br /> 1) x_- = x_+<br />$

$<br /> 2) \frac{(-b + \sqrt{ b^2 - 4ac } )}{2} = \frac{(-b - \sqrt{ b^2 - 4ac } )}{2}<br />$

$<br /> 3) -b + \sqrt{ b^2 - 4ac } = -b - \sqrt{ b^2 - 4ac }<br />$

$<br /> 4) \sqrt{ b^2 - 4ac } = - \sqrt{ b^2 - 4ac }<br />$

$<br /> 5) b^2 -4ac = -b^2 + 4ac<br />$

$<br /> 7) 2b^2=8ac<br />$

$<br /> 8) c = \frac{2b^2} {8a}<br />$

$<br /> 9) c = \frac{b^2} {4a}<br />$

and that actually works. For any quadratic equation with the given values a and b i can find a c such that the equation only has one solution. The problem is, in retrospect i have no idea why this is true because i completely fucked up that derivation. Mainly in step 5 where i raised both sides to the power of 2 to remove the square roots, but i forgot that, that should also makes the "-" disappear from the right side and so it should just equal itself and be unsolvable from there. However because i forgot about it i ended up deriving something that is still true? how can that be?

also before that in step 4 i have sqrt(something) = - sqrt(something), how can that be true? as far as i can tell i didn't do anything wrong algebraically up until then, and the initial equation should be true right?

condensed matter focused undergrad here; are there any amo physicists on this board? thinking about switching to that subfield in graduate school cause the day to day stuff seems cooler but I have absolutely no idea how anything you guys do is important.

Do you think that Einstein was overrated? And why?

How do you prove this is true for every combination of x and y? All variables in the equation are integers.

And in what context is it meaningful?
The number googolplex is only significant as a thought experiment and as an etymological explanation for the meaning of Google.
So what is the cutoff where, beyond that point, the numbers are practically worthless?

>want to study neuroscience
>can't justify it seeing as how there are few if any careers where the financial payoff is greater than the student debt I'd be amassing

Why can't the market favor neurofags?

Are supplements like these just placebo?

Constructively judge and critique my paper on The Continuum Hypothesis.