# Expressing a quadratic form in canonical form using Lagrange

• Apothem
In summary, by completing the square, you can express the given quadratic form in canonical form using Lagrange's Method/Algorithm.
Apothem
Moved from technical forums, so no template
Problem:

q=x1x2+x1x3+x2x3

in canonical form using Lagrange's Method/Algorithm

Attempt:
Not really applicable in this case due to the nature of my question

Using the change of variables:
x1=y1+y2
x2=y1-y2
x3=y3
Indeed you get q'=(y1+y3)2-(y2)2-(y3)2

I can see that when you substitute it in it does indeed reduce it to canonical form however my question is as follows:
How am I meant to arrive at the above change of variables? I have seen a few cases of completing the square, but to my knowledge we can't use that here. EDIT: We can

Any help is greatly appreciated!

EDIT:
I think I've been able to deduce it...
x1x2+x1x3+x2x3=(x3)2+x3(x2+x1)-(x3)2 and from there I completed the square, the only thing I'm still a bit uncertain on is that I worked this out the other way i.e. I worked out y in terms of x and not x in terms of y...

Last edited:
Is this ok? Yes, that is fine. Completing the square is a valid approach to solving this problem and it is essentially the same as Lagrange's Method/Algorithm. By completing the square, you are taking a quadratic expression of the form ax2 + bx + c and rewriting it as (x - x0)2 + d, where x0 is the vertex of the parabola. That is essentially what you are doing in your solution, and it is equivalent to using Lagrange's method.

## 1. What is a quadratic form?

A quadratic form is a mathematical expression that contains variables raised to the second power, such as x^2 or y^2. It can also include variables multiplied together, such as xy or x^2y.

## 2. What is canonical form?

Canonical form is a standard or preferred way of expressing a mathematical expression or equation. In the case of quadratic forms, canonical form is a specific way of writing the expression that makes it easier to analyze and solve.

## 3. How do you express a quadratic form in canonical form?

To express a quadratic form in canonical form, you can use the method of completing the square or the Lagrange method. The Lagrange method involves using a matrix transformation to simplify the expression and make it easier to identify the key elements of the quadratic form.

## 4. What is the Lagrange method?

The Lagrange method is a mathematical technique used to simplify and analyze quadratic forms. It involves using a matrix transformation to transform the quadratic form into a simpler expression, which can then be written in canonical form.

## 5. Why is it useful to express a quadratic form in canonical form using Lagrange?

Expressing a quadratic form in canonical form using Lagrange can make it easier to analyze and solve the expression. It can also help to identify important characteristics of the quadratic form, such as the direction of the quadratic form, the location of the minimum or maximum value, and the relationship between the variables in the expression.

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