Deriving the Quadratic Formula

You can optionally see also https://www.youtube.com/watch?v=OZNHYZXbLY8 and http://www.youtube.com/watch?v=bjH1HphOZ1Y for what is completing square.

I will show alternative way to accomplish this task.

(a+b)²=a²+2ab+b²

So, the tough question will be

x²+3x+4=2

x²+2*(3/2)*x+4=2

x²+2*(3/2)*x+(9/4)–(9/4)+4=2

(x+(3/2))²=2–4+(9/4)

(x+(3/2))²=(1/4)

x+(3/2)=(1/2) or x+(3/2)=-(1/2)

x=-1 or x=-2

And so, alternative derivation would be:

We want to solve

ax²+bx+c=0, where a≠0

Because a≠0 we can divide both parts by a, we will get

0=x²+(b/a)x+(c/a)=x²+2*(b/2a)*x+(c/a)=

=x²+2*(b/2a)*x+(b²/4a²)-(b²/4a²)+(c/a)=

=(x+(b/2a))²-(b²/4a²)+(c/a)

or another way around

(x+(b/2a))²=(b²/4a²)-(c/a)

(x+(b/2a))²=(b²/4a²)-(4ac/4a²)

(x+(b/2a))²=(b²-4ac)/4a²

(x+(b/2a))=± sqrt((b²-4ac)/4a²)

From here:

(x+(b/2a))=± (sqrt(b²-4ac)/2a)

x=-(b/2a)± (sqrt(b²-4ac)/2a)

After, you know the formula, solving the tough question will be much easier:

x²+3x+4=2

x²+3x+2=0

Originally published in my blog https://www.toalexsmail.com/2012/04/deriving-quadratic-formula-english.html

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