Computer is dying

My desktop is having problems booting up. I guess the hardware inside my computer is really failing. I'm not sure when it will finally crumble. And today, my BIOS gone haywire at me, but in the end, the RESET button did all the work. Yeah, so it finally booted up. Even so, the computer crashed, as usual. Two nights ago, my desktop had more unusual crashes. I have never seen those messages before in the Blue Screen. Now, that really freaked me out. Even if I could search up Microsoft about the Bug check codes that were displayed, how could it help me? And, sending the bug check information to Microsoft doesn't really do much good also. All it could say is "a driver encountered a problem" or something similar. Yeah, like that's gonna help.

Anyway,

Here are some more maths! Sorry, I'm just lazy to write it out in my Nonsense Book, so I'll just type it out. I found this in my sister's Math book and Wikipedia. It's about the Quadratics formula and how it came to be.

I had used this formula in my post about Circle-Line intersection:


Okay, the question is, how did it turn from this:
ax²+bx+c=0

to that?

Before I get started with the OMG-IT'S-SO-UNBELIEVABLE maths, let's talk about completing the squares.

Huh? What? Completing the squares?

Anyway, completing a square is just a method used to solve quadratic equations. Right, imagine you have an expression, x² + 2xy. We want to factorise this so that it'll be in the form of (x+y)². Before we can do that, we have to find out what's missing in x² + 2xy that doesn't allow it to be factorised into the (x+y)². Still with me so far?

Let's expand (x+y)²:
x²+2xy+y²

As you can see, the missing term is y². Just add the y² into the x² + 2xy and it will be factorisable into the (x+y)². Hmm... factorisable, is there such a word?

Let's take an example:
x²+6x

We need to find the "y". We know that
2xy=6x

So,
y=3;

And, y²=9

Therefore, it'll be x²+6x+9. This is then factorised into (x+3)².

So, did you get it? I hope you did because we are gonna apply that later in the post.

Okay, now we know how to complete a square. Let's go back to the
ax²+bx+c=0

At first glance, this is cannot be factorised, because a,b and c are not known. But, we could apply what we've learnt just now.

First, we need to make the left side of the equation to be in the form of x²+2xy. To do that,

ax²+bx+c=0
ax²+bx=-c

Now divide the whole thing with "a"
x²+ (b/a) x=-c/a

Good. Now, we can say that,
2xy=(b/a) x
So, y=(b/2a)
y²= (b²/4a²)

Add that into the equation. Remember, whatever you do to one side of the equation must be done on the other side as well. That's the basic rule of algebra.
x² + (b/a) x + (b²/4a²)=-c/a + (b²/4a²)

Factorise the left side of the equation and add up the terms in the second.
And you'll get:

[x+(b/2a)]²=[ (b²-4ac)/4a² ]

Square root everything

x+(b/2a) =[ sqrt(b²-4ac) ] / 2a

Solve for x

x = [ sqrt(b²-4ac) ] / 2a - (b/2a)
x = [-b ± sqrt(b²-4ac)]/2a

And that's how you get:


Hope you liked it!! I know I did

Here's a better way to see it, courtesy of Wikipedia: