Question

Solve this question with explanation​

Answer 1

Step-by-step explanation:

hope it helps all the best

Answer 2

Here, we will use simple concepts, which, just for the record, come under the name of Surds.

The question is a complete square root. However, the options don't consist of a square root sign on the whole expression. This hints that some kind of perfect square must be forming.

The solution goes like this:

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

Here, we observe this: $2x\sqrt{x^2-1}$

If a perfect square must be forming, this is a worthy middle term.

Basically, if we are getting something like $\sqrt{a^2+2ab+b^2}$, then this $2ab$ corresponds to $2x\sqrt{x^2-1}$

Also, a little careful observation shows that if we compare $2x\sqrt{x^2-1}$ with $2ab$, then there must also be the equivalent expressions of $a^2$ and $b^2$ in the original question.

And, this indeed is the case. Here's how:

$\sqrt{2x^2-1+2x\sqrt{x^2-1}}\\\\\\=\sqrt{x^2+x^2-1+2x\sqrt{x^2-1}}\\\\\\=\sqrt{(x)^2+2x\sqrt{x^2-1}+\left(\sqrt{x^2-1}\right)^2}\\\\\\\left[\quad\equiv\sqrt{a^2+2ab+b^2}\quad\right]\\\\\\=\sqrt{\left(x+\sqrt{x^2-1}\right)^2}\\\\\\=x+\sqrt{x^2-1}$

And so we have:

$\boxed{\bold{\sqrt{2x^2-1+2x\sqrt{x^2-1}}=x+\sqrt{x^2-1}}}$

Thus, The Answer is Option (2) $\bold{x+\sqrt{x^2-1}}$