Putting an equation in quadratic form

In summary, to put an equation into quadratic form, you need to first make a square for the x terms and another square for the y terms without touching the x. Then, use the z^2 term to balance the z's. To do this, choose numbers c and a such that you get the correct x^2 and xy terms. Multiply everything out and add the appropriate amount of z^2 at the end to complete the quadratic form.
  • #1
miahmad
5
0
Could someone please explain how to go about putting an equation into quadratic form.
e.g:
Q(x,y,z)=7x^2-2y^2-40z^2-14xz+20yz.

I know it equals 7(x-z)^2 -2(y-5z)^2 +3z^2. dnt know how to get there though.
 
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  • #2
Dnt wrry figured it out.
 
  • #3
What you want to do is first make a square which gets all the x terms right, then a square which gets all the y terms right, but without touching the x. Then in the end, you can use the z2 term to balance the z's.

So, for example, with an x you have 7x2 and -14xz.
If you write c (x - a)2, you will get terms with c x2 and 2 x a c, and stuff with a2, which you can cancel by -a2 terms, if you need. So you can reproduce these two terms by taking c = 7, a = -z.

Then you do the same with the y, choosing numbers c and a such that you exactly get the right y2 and yz terms.

Then, you multiply out everything you have so far. If you did it right, all the terms will match, except possibly those with only z2. Since z2 is itself a square, you can easily fix this by adding the appropriate amount of z2 at the end, and you're done.
 

1. What is quadratic form?

Quadratic form is a mathematical expression that can be written in the form of ax^2 + bx + c, where a, b, and c are constants and x is the variable. It is commonly used to solve for the roots or solutions of a quadratic equation.

2. Why is it important to put an equation in quadratic form?

Putting an equation in quadratic form makes it easier to solve for the roots or solutions of the equation. It also helps in graphing the equation and understanding its behavior.

3. How do you put an equation in quadratic form?

To put an equation in quadratic form, you need to make sure that the highest power of the variable is 2. If it is not, you can use algebraic techniques such as completing the square or the quadratic formula to transform the equation into quadratic form.

4. What are some real-life applications of quadratic form?

Quadratic form is used in various fields such as physics, engineering, and statistics. It can be used to model the trajectory of a projectile, determine the optimal shape of an object, or analyze data in regression analysis.

5. Can an equation be in quadratic form if it only has one term?

No, an equation must have at least two terms, one with a variable raised to the power of 2 and one without, to be in quadratic form. If an equation only has one term, it is considered linear and can be written as ax + b, where a and b are constants and x is the variable.

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