Quadratic Equation: Who Invented & What's the Proof?

AI Thread Summary
The discussion centers around the origins and proof of the quadratic equation, which is derived by completing the square on the general form ax^2 + bx + c = 0. While the exact inventor is unclear, it is suggested that an Arabic mathematician may have first proved it. A recent article in the Mathematics Teacher presents an alternative, elegant method to derive the quadratic formula, focusing on the roots and vertex of the parabola. This method highlights the relationship between the coefficients and the roots, leading to a visual representation of the equation. The conversation emphasizes the beauty and complexity of mathematical proofs related to the quadratic equation.
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who came up with the Quadratic equation and what is the proof behind it?
 
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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.
 
Quantumduck said:
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?
 
thharrimw said:
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.
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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.
 
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what is ^?
 
  • #10
Ephratah7 said:
what is ^?

Exponentiation. It means "raised to the power of"
 

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