Question

solve in detail.
Easiest solution will be marked as brainliest

Answer 1

Thanks for asking the question!

ANSWER::

I am going to show you a very lazy method as you have given me options too. So , let's start

Question = √[2x² -1 + 2x√(x²-1)]

Let us check the answer with every option::

1. Suppose option 1 is the answer

√[2x² -1 + 2x√(x²-1)] = x - √(x²-1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² - 1 - 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² -1 - 2x√(x²-1)

LHS is not equal to RHS so , 1st option is not the answer.

2. Suppose option 2 is the answer

√[2x² -1 + 2x√(x²-1)] = x + √(x²-1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² - 1 + 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² -1 + 2x√(x²-1)

LHS is equal to RHS so , option 2 is correct.

3. Suppose option 3 is the answer

√[2x² -1 + 2x√(x²-1)] = x + √(x²+1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² + 1 + 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² +1 + 2x√(x²-1)

LHS is not equal to RHS so , option 3 is not the answer.

4. Suppose option 4 is the answer

√[2x² -1 + 2x√(x²-1)] = x - √(x²+1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² + 1 - 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² + 1 - 2x√(x²-1)

LHS is not equal to RHS so , option 4 is not the answer.

Basically , as we were applying option 2nd option came to be correct so , we can leave the other options and not check them if its a single choice question.

But , if it is a Multiple Choice question we have to check all options.

Hope it helps!

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