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

In summary, the Quadratic equation was first derived by an Arabic mathematician and can be proven using various methods such as completing the square or through visual methods. The latest method, published by the NCTM, involves finding the roots and vertex of the equation and using algebra to solve for x. The exponentiation symbol, ^, represents raising a number to a certain power.
  • #1
thharrimw
115
0
who came up with the Quadratic equation and what is the proof behind it?
 
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  • #3
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.
 
  • #4
Take ax^2+bx+c=0 complete the square and you'll obtain the quadratic equation.
 
  • #5
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.
 
  • #6
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?
 
  • #7
thharrimw said:
what is it?

Yeah I'm curious too
 
  • #8
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.
 
Last edited:
  • #9
what is ^?
 
  • #10
Ephratah7 said:
what is ^?

Exponentiation. It means "raised to the power of"
 

1. Who invented the quadratic equation?

The concept of the quadratic equation can be traced back to ancient Babylonian and Greek mathematicians, but the first known systematic solution was provided by Indian mathematician Brahmagupta in the 7th century. The modern form of the quadratic equation was developed by Persian mathematician Al-Khwarizmi in the 9th century.

2. What is the proof for the quadratic equation?

The proof for the quadratic equation relies on basic algebraic principles and the use of the quadratic formula. It starts with the general form of a quadratic equation, ax^2 + bx + c = 0, and then uses the quadratic formula to solve for the roots. The proof can be found in most high school and college level algebra textbooks.

3. Why is the quadratic equation important?

The quadratic equation is important because it is used to solve a wide range of real-world problems, from finding the maximum or minimum value of a function to predicting the trajectory of a projectile. It is also a fundamental concept in algebra and is used in higher level math courses such as calculus.

4. Can the quadratic equation be used to solve all polynomial equations?

No, the quadratic equation can only be used to solve equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Polynomial equations of higher degrees (such as cubic or quartic equations) require different methods to solve.

5. What are the applications of the quadratic equation in the real world?

The quadratic equation has many practical applications in fields such as physics, engineering, economics, and statistics. It can be used to model and solve problems involving motion, optimization, and probability. Examples include calculating the maximum height of a ball thrown into the air, determining the most profitable production level for a company, and predicting the likelihood of a certain event occurring.

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