1. ## a rational number?

Hello. I'd like to ask if there is any possibility that
sqrt(7) - sqrt(x)
can give a rational number?

2. Originally Posted by olaolaola
Hello. I'd like to ask if there is any possibility that
sqrt(7) - sqrt(x)
can give a rational number?
Hint: 0 is rational

3. yes, i meant except of 0?

4. Originally Posted by olaolaola
yes, i meant except of 0?
Solve $\displaystyle \sqrt{7}-\sqrt{x}=1$ for x.

5. Yes, I see. But in this equation x is not a rational number. And if in my equation: sqrt(7) - sqrt(x) , x is a rational number? Then can the result be somehow rational (and not 0)?

6. Originally Posted by olaolaola
Yes, I see. But in this equation x is not a rational number. And if in my equation: sqrt(7) - sqrt(x) , x is a rational number? Then can the result be somehow rational (and not 0)?
You do understand this wastes people's time, if you don't specify the question clearly and we end up answering the wrong question?

Let $\displaystyle a,b\in\mathbb{Z},b\ne0$

$\displaystyle \sqrt{7}-\sqrt{x}=\frac{a}{b}$

$\displaystyle \sqrt{7}-\frac{a}{b}=\sqrt{x}$

$\displaystyle x=(\sqrt{7}-\frac{a}{b})^2$

$\displaystyle x = 7-\left(\frac{2a}{b}\right)\sqrt{7}+\frac{a^2}{b^2}$

With a little more work we see that the only way to make x rational is to let a = 0.

7. Yes, I understand, next time I promise I will be more precise. But I'm very grateful, you helped me very much. Thank you!

8. x=7