Integer solutions to a function

This is from a past iGCSE paper. Expanding and re-arranging the equation is fairly easy, but except by trial and error, or plotting it, ow can you find the integer solution for x and y?


Given that :

(5-sqrt(x))^2 = y - 20*sqrt(2)

where x and y are positive integers, find the value of x and the value of y.


---Thread moved to Maths---

What was the full question? Do you just need to find one set of x and y that satisfy the equation?

If you expand you get:

$25-10\sqrt{x}+x = y-20\sqrt{2}$

Then if you try to match up the surd term on either side you get

$-10\sqrt{x} = -20\sqrt{2}$

Can you see what $x$ must be to satisfy this? You can compare terms in the rest of the equation to get $y$ once you've got $x$.

Please post all your working if you get stuck.

Given that :

$(5-\sqrt{x})^2 = y - 20\sqrt{2}$

where x and y are positive integers, find the value of x and the value of y .

Thanks for the reply. The full question was what I quoted in the first post.



I'm sorry for being stupid, but what do you mean by "match up the surds on either side"? I can see that $\sqrt{2}$ is a surd, but how do you know that $\sqrt{x}$ is? Also, why does this matching up eliminate $y$? Confused....

Ah, thank you. Yes, I understand that. The key is to split the equation into two separate equalities - very neat . The question is from the Pearson Edexcel Jan 14 3H paper:

http://qualifications.pearson.com/content/dam/pdf/Edexcel%20Certificate/Mathematics%20A/2009/Exam%20materials/4MA0_3H_que_20140110.pdf

Question 17.

Thanks again - much appreciated!

$-10\sqrt{x} + (25+x) = -20\sqrt{2} + (y)$


Intead of rearranging to solve this like what you're used to in algebra, we're going to find a solution by matching up both sides of the equation. So what we need are the two surd terms to be the same

i.e. $-10\sqrt{x} = -20\sqrt{2}$

And we need the rest to be the same

$25+x = y$

Actually, on reviewing this, I do have another question....

Can you explain how the following works...




You have simply said "we need the two surd terms to be the same", but I'm not sure why, or how it's valid to do that. Being stupid for the moment, I could have this:

$5 + 3 = 7 + 1$

But I can't just arbitrarily convert this into:

$5 = 7$ and $3 = 1$

So, why is it OK to match up the surd terms?

Sorry for more dumb questions...

Good question and I'm going to have to ramble a bit to explain it


If you have an equation with more than one unknown in it e.g.

$x+2y^2 + 3x = 5y - 2 + x$

then let's say you group some terms so you end up with something like this:

(stuff) + (more stuff) = (blah) + (more blah)

e.g. "stuff" could be $x+2y^2$, which would make "more stuff" $3x$.

If you can find values for your variables such that "stuff = blah" and "more stuff = more blah" then it makes sense that these values must be solutions to your equation since the equation would be balanced.

But this technique isn't guaranteed to work and most of the time it won't help you solve the equation because the values that satisfy "stuff" = "blah" won't also satisfy "more stuff" = "more blah".

Also this tecnhique doesn't always work the other way around i.e. when you've already got a solved equation. So if you have something like

$5 + 3 = 7 + 1$

then it's not necessarily the case that the two left terms (5 and 7) are equal and the two right terms are equal, and you have already shown this yourself.


Now going back to this equation:

$-10\sqrt{x} + (25+x) = -20\sqrt{2} + (y)$

The reason this technique was so useful is because solving the surd part gives you a single integer value for $x$, which you can plug in to the other equation to give you an integer value for $y$.

Also (and maybe more importantly), $-20\sqrt{2}$ is irrational - let me know if you haven't heard of this word. So since $y$ and $x$ are integers, the only possible other term that can equate with $-20\sqrt{2}$ to balance the equation is $-10\sqrt{x}$.

So when you have an equation that has rational and irrational parts, a useful technique is to equate the rational parts and then equate the irrational parts.

...since $y$ and $x$ are integers, the only possible other term that can equate with $-20\sqrt{2}$ to balance the equation is $-10\sqrt{x}$.