# Thread: Perfect square root of a quadratic

1. ## Perfect square root of a quadratic

Is it possible to find analytically the function that forms the perfect square root of a quadratic in all cases?

For example with the quadratic x^2 + 10x + 25 you can find by factoring the perfect root x + 5. But how do you find the square root when a quadratic isn't expressed in the form of the completed square?

With the quadratic x^2 + 10x + 20 it seems the simplest form you can get the root into is sqrt((x+5)^2 -5)).

Is that the best that can be done? Is there some other way to express this root?

2. I really cannot make heads or tails out of what you are saying. Either a quadratic is a perfect square or it is not. You cannot change a quadratic that is not a perfect square to a different "form" in which it is a perfect square. And I have no idea why you would prefer $\displaystyle \sqrt{(x+ 5)^2- 5}$ to simply $\displaystyle \sqrt{x^2+ 10x+ 20}$. They both are "the function that forms the perfect square root of a quadratic".

3. Originally Posted by HallsofIvy
I really cannot make heads or tails out of what you are saying. Either a quadratic is a perfect square or it is not. You cannot change a quadratic that is not a perfect square to a different "form" in which it is a perfect square. And I have no idea why you would prefer $\displaystyle \sqrt{(x+ 5)^2- 5}$ to simply $\displaystyle \sqrt{x^2+ 10x+ 20}$. They both are "the function that forms the perfect square root of a quadratic".
I see. I ask such a strange question because I have a calculation in which a square rooted quadratic results for each step and x is unknown at the time of the calculation. It seems I will have to store the complete series eg. $\displaystyle \sqrt{5.123x^2 + 10.235x + 20.8} + \sqrt{2x^2 + 4.234x + 13.35} + ...$ whereas I was mistakenly thinking I would be able to simplify terms together. Thanks for your reply.