Completing the square involving square roots

[Monsterman222]

Hi, I was looking at the derivation of the equation for a hyperbola on Wolfram Mathworld. In one step, the webpage instructs you to "complete the square". It starts with:

$$\sqrt{\left(x -c\right)^{2} +y^{2}}-\sqrt{\left(x+c\right)^{2}+y^{2}} = 2a$$

and then says, "rearranging and completing the square gives":

$$x^{2}\left(c^{2}-a^{2}\right)-a^{2}y^{2}=a^{2}\left(c^{2}-a^{2}\right)$$

How did he do this? The original page can be found at http://mathworld.wolfram.com/Hyperbola.html and it's equations (3) and (4).

 

[tiny-tim]

Hi Monsterman222!

It's wrong …

it's not completing the square, it's just squaring both sides (after rearranging), twice

 

[ardie]

you do have to complete the square eventually, just do square both sides frist

 

[Monsterman222]

Awesome, thanks! I was able to get it without completing the square, just squaring twice qfter rearranging.

 

[CellCoree]

Hello,

Completing the square is a method used to rewrite a quadratic equation in a specific form. In this case, the equation is not in a standard quadratic form, but by rearranging and completing the square, it can be rewritten as a standard quadratic equation.

To complete the square, we first need to isolate the square root terms on one side of the equation. In this case, we can bring the second square root term to the other side by adding it to both sides of the equation:

\sqrt{\left(x -c\right)^{2} +y^{2}} = 2a + \sqrt{\left(x+c\right)^{2}+y^{2}}

Next, we can square both sides of the equation to get rid of the square root:

\left(x -c\right)^{2} +y^{2} = \left(2a + \sqrt{\left(x+c\right)^{2}+y^{2}}\right)^{2}

Expanding the right side of the equation, we get:

\left(x -c\right)^{2} +y^{2} = 4a^{2} + 4a\sqrt{\left(x+c\right)^{2}+y^{2}} + \left(x+c\right)^{2}+y^{2}

Now, we can rearrange the terms to group the square terms together and the non-square terms together:

\left(x -c\right)^{2}+\left(x+c\right)^{2} +y^{2}+y^{2} = 4a^{2} + 4a\sqrt{\left(x+c\right)^{2}+y^{2}}

Simplifying, we get:

2x^{2}+2c^{2}+2y^{2} = 4a^{2} + 4a\sqrt{\left(x+c\right)^{2}+y^{2}}

Now, we can divide both sides by 2 to get:

x^{2}+c^{2}+y^{2} = 2a^{2} + 2a\sqrt{\left(x+c\right)^{2}+y^{2}}

Finally, we can subtract 2a^{2} from both sides to get the equation in the form given on the Mathworld webpage:

x^{2}\left(c^{2