# Thread: Need help understanding how to factor out of a complex root

1. ## Need help understanding how to factor out of a complex root

In the process of brushing up on my math skills, I came across an example where $\sqrt{x^2+x}+x$ has $x$ factored out, resulting in $x(\sqrt{1+\frac{1}{x}}+1)$.

I am confused as to how this is accomplished, and would greatly appreciate it if someone could enlighten me.

I tried looking over various texts on dealing with exponents, but none of them cover this type of issue.

2. $\sqrt{x^2+x}+x$

$\sqrt{x(x+1)}+x$

$\sqrt{x}\sqrt{x+1}+x$

$x^{\frac{1}{2}}\sqrt{x+1}+x$

$x\times x^{\frac{-1}{2}}\sqrt{x+1}+x$

$x\left( x^{\frac{-1}{2}}\sqrt{x+1}+1\right)$

You can simplify the expression inside the brackets to finish.

3. Originally Posted by Lancet
In the process of brushing up on my math skills, I came across an example where $\sqrt{x^2+x}+x$ has $x$ factored out, resulting in $x(\sqrt{1+\frac{1}{x}}+1)$.

I am confused as to how this is accomplished, and would greatly appreciate it if someone could enlighten me.

I tried looking over various texts on dealing with exponents, but none of them cover this type of issue.
for $x > 0$ ...

$\sqrt{x^2+x} + x =$

$\displaystyle \sqrt{x^2\left(1+\frac{1}{x}\right)} + x =$

$\displaystyle \sqrt{x^2} \cdot \sqrt{1+\frac{1}{x}} + x =$

$\displaystyle x \cdot \sqrt{1+\frac{1}{x}} + x =$

$\displaystyle x\left(\sqrt{1+\frac{1}{x}} + 1\right)$

4. Thank you both. I see where the missing techniques are.