# Thread: How to extract square root of three or more terms?

1. ## How to extract square root of three or more terms?

Qu: Dear Math helpers, I have been tackling a book by Durell called School Algebra to improve my math skills and to improve my algebraic manipulation skills.
I have learn how to square trinomial such as (a+b+c)(a+b+c) etc. However, I have hit a bump.

I have shared the link of the book below and a picture of page. The place where i am stuck is page 283. The problem is that I don't understand the method and the example given by author.
The example was: Extact the square root of $x^4 - 6x^3 + 19x^2 - 30x + 25$

Thank you ver much for your help

2. ## Re: How to extract square root of three or more terms?

I'm not familiar with the author's process ... I'd have to go over it if I have time.

I would approach the problem in this manner ...

If it is reducible, the quartic polynomial will have a quadratic square root, i.e. ...

$x^4-6x^3+19x^2-30x+25 = (ax^2+bx+c)^2$

$x^4-6x^3+19x^2-30x+25 = a^2x^2 + 2abx^3 + (2ac+b^2)x^2 + 2bcx + c^2$

matching coefficients, it's easy to see $a=1$ and $c=5$

$2b = -6 \implies b = -3$

so ...

$(x^2-3x+5)^2 = x^4-6x^3+19x^2-30x+25$

now, would $(-x^2 + 3x -5)^2$ also work?

3. ## Re: How to extract square root of three or more terms?

Originally Posted by skeeter
I'm not familiar with the author's process ... I'd have to go over it if I have time.

I would approach the problem in this manner ...

If it is reducible, the quartic polynomial will have a quadratic square root, i.e. ...

$x^4-6x^3+19x^2-30x+25 = (ax^2+bx+c)^2$

$x^4-6x^3+19x^2-30x+25 = a^2x^2 + 2abx^3 + (2ac+b^2)x^2 + 2bcx + c^2$

matching coefficients, it's easy to see $a=1$ and $c=5$

$2b = -6 \implies b = -3$

so ...

$(x^2-3x+5)^2 = x^4-6x^3+19x^2-30x+25$

now, would $(-x^2 + 3x -5)^2$ also work?
thank you very much