Completing the square 2 variables

In summary, the given equation is being used in a book to demonstrate completing the square, but the author does it in one line without showing the procedure. The problem is solved by using the known formula (u+v)^2 = u^2 + 2uv + v^2 and identifying u and v as \sqrt{A}x and \sqrt{B}y respectively. This leads to a solution of (\sqrt{A}x + \sqrt{B}y)^2 - (Mxy + 2\sqrt{AB}xy).
  • #1
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Homework Statement


Ax[itex]^{2}[/itex]+By[itex]^{2}[/itex]-Mxy

Capitals are constants.

The equation (one like it) is in a book I am reading where the author procedes to complete the square. He does it in one line without showing the procedure and I am completely baffled.

Homework Equations





The Attempt at a Solution



 
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  • #2
Well, what exactly is the problem? Do you know how to complete the square?
I will presume that you at least know that [itex](u+ v)^2= u^2+ 2uv+ v^2[/itex].
Since you have [itex]Ax^2[/itex] and [itex]By^2[/itex] as [itex]u^2[/itex] and [itex]v^2[/itex], you must have [itex]u= \sqrt{A}x[/itex] and [itex]v= \sqrt{B}y[/itex] so that [itex]2uv= 2\sqrt{AB}xy[/itex].

That is, we can write
[tex]Ax^2+B^2- Mxy= Ax^2+ 2\sqrt{AB}xy+ By^2- Mxy- 2\sqrt{AB}xy[/tex]
[tex]= (\sqrt{A}x+ \sqrt{B}y)^2- (Mxy+ 2\sqrt{AB}xy)[/tex]
 
  • #3
Here it is. I am pleased to say it all makes sense now.
 

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Related to Completing the square 2 variables

1. What is the purpose of completing the square with 2 variables?

The purpose of completing the square with 2 variables is to transform a quadratic equation into its standard form, which can be easier to work with and solve. This process is also helpful in graphing the equation and understanding its properties.

2. What are the steps for completing the square with 2 variables?

The steps for completing the square with 2 variables are:
1. Write the equation in the form ax^2 + bx + c = 0.
2. Move the constant term (c) to the right side of the equation.
3. Divide the coefficient of x^2 (a) by 2 and square the result.
4. Add the squared result to both sides of the equation.
5. Factor the left side of the equation, and simplify the right side.
6. Take the square root of both sides of the equation.
7. Solve for x by adding or subtracting the constant term (b/2a) from both sides.

3. Why is it important to complete the square in 2 variables?

Completing the square in 2 variables is important because it helps us to find the vertex of the parabola represented by the quadratic equation. This vertex is the maximum or minimum point of the parabola and provides important information about the equation and its solutions.

4. Can completing the square be used to solve any quadratic equation with 2 variables?

Yes, completing the square can be used to solve any quadratic equation with 2 variables. It is a reliable method for solving equations that cannot be easily factored or solved using other methods, such as the quadratic formula.

5. Are there any shortcuts for completing the square with 2 variables?

Yes, there are some shortcuts for completing the square with 2 variables, such as using the quadratic formula or using a graphing calculator. However, these shortcuts may not always be applicable or may not provide a complete understanding of the process, so it is important to understand the steps and principles behind completing the square.

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