# Completing the square involving square roots

• Monsterman222
In summary, the derivation of the equation for a hyperbola involves squaring both sides and rearranging the terms. Completing the square is not necessary, but it can be done eventually. The webpage on Wolfram Mathworld provides equations (3) and (4) as an example of this process.
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).

Hi Monsterman222!

It's wrong

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

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

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

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

## 1. What is completing the square involving square roots?

Completing the square involving square roots is a mathematical method used to solve quadratic equations that involve square roots. It involves manipulating the equation to express it in a perfect square form.

## 2. Why is completing the square involving square roots useful?

This method is useful because it allows us to solve quadratic equations that cannot be solved by factoring or using the quadratic formula. It also helps us find the vertex of a parabola.

## 3. How do you complete the square involving square roots?

To complete the square involving square roots, we follow these steps:
1. Move the constant term to the right side of the equation.
2. Divide the coefficient of the x term by 2 and square it.
3. Add the squared value to both sides of the equation.
4. Factor the perfect square on the left side of the equation.
5. Simplify and solve for x by taking the square root of both sides of the equation.

## 4. Can completing the square involving square roots be used for all quadratic equations?

Yes, completing the square involving square roots can be used for all quadratic equations. However, it is most useful for equations where the coefficient of the x^2 term is not equal to 1.

## 5. How do you know if you have completed the square involving square roots correctly?

You can check if you have completed the square involving square roots correctly by multiplying out the perfect square on the left side of the equation. The result should be equal to the original equation. Additionally, the solutions obtained from completing the square should be the same as the solutions obtained from using the quadratic formula.

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