who came up with the Quadratic equation and what is the proof behind it?
For the proof have a look at http://mathworld.wolfram.com/QuadraticEquation.html
and for more details at http://en.wikipedia.org/wiki/Quadratic_equation
It's just completing the square, but used on the general form of the quadratic equation. I can't remember who first proved it though :( Some Arabic mathematician i think.
Take ax^2+bx+c=0 complete the square and you'll obtain the quadratic equation.
The latest edition of the Mathematics Teacher, published by the NCTM has a different way to derive the quadratic formula than completing the square.
It is actually kind of elegant as well.
what is it?
Yeah I'm curious too
Dang, I knew I should have brought my copy home. I will look it up and post later.
Ok, looked it up. It is complex, so I will have to short cut it. The full citation, if interested is:
Piccioto, Henry. (February 2008). A new path to the quadratic formula. Mathematics Teacher 101:6, 473-478.
If there are roots p and q, then the function can be written in factored form
y=a(x-p)(x-q) = x^2 - a(p+q)x + apq
It follows that the product of the roots is c/a, since c=apq and the sum of the roots is -b/a, since b= -a(p+q).
From here, he uses that information to find (h,v), the co-ordinates of the vertex. The average of the roots, h, is -b/2a. This is then substituted into the formula to get v, and the resultant is
v= (-b^2 +4ac)/4a
Notice that this is the discriminant divided by 4a!
Finally, the author notes that the x intercept is on either side of the vertex by the same amount, d, so x = -b/2a +- d, and if we move the parabola so that the vertex is at the origin, it's equation simply becomes y=ax^2.
With this new translated parabola, we can then do a little algebra (which is explained in the article, 2 steps) to get x = the negative boy couldn't decide on whether to attend a radical party or be square, so he missed out on 4 awesome chicks and the party was all over by 2 am.
It is a very visual method, instead of the normal completing the square method.
There is no way I did it justice in my re-telling.
what is ^?
Exponentiation. It means "raised to the power of"
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