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Confused on surds question

Decided to go back and do some harder questions in my A-Level textbook and one question is asking me find the values of x and y for the equation (5√7 - √x)/(√7 - √x)= (8 + √y)
The answers are x is 3 and y is 21, but I arrived at this by plugging digits into x and seeing which equation works.
(edited 2 months ago)
Reply 1
Original post by Sorbus25
Decided to go back and do some harder questions in my A-Level textbook and one question is asking me find the values of x and y for the equation 5√7- √x/√7 - √x= 8 + √y.
The answers are x is 3 and y is 21, but I arrived at this by plugging digits into x and seeing which equation works.

You could have included some brackets but Im taking it as
(5sqrt(7)-sqrt(x))/(sqrt(7)-sqrt(x))=8+sqrt(y)
and youre looking for integer solutions, otherwise its just a curve. You could divide out the left hand side to get quotient+remainder/divisor and get the domain of x, or take the 8 over and reason about the resultant fraction which is probably the easiest way to see which (restricted) value of x is going to give.

You could also try to rationalise the left hand side and square up as necessary, but the previous approach would be the first thing to try.
(edited 2 months ago)
Reply 2
Original post by mqb2766
You could have included some brackets but Im taking it as
(5sqrt(7)-sqrt(x))/(sqrt(7)-sqrt(x))=8+sqrt(y)
and youre looking for integer solutions, otherwise its just a curve. Probably the first thing to try is to divide out the left hand side to get quotient+remainder/divisor and reason about that,but not worked it through fully. The sqrt(7) on the numerator and denominator is a bit of a hint it may give.

You could also try to rationalise the left hand side and square up as necessary, but the previous approach would be the first thing to try.

Thank you. My concern is that I thought that an equation with two unknown variables cannot be solved. Unless I treat the equation as a graph?
Reply 3
Original post by Sorbus25
Thank you. My concern is that I thought that an equation with two unknown variables cannot be solved. Unless I treat the equation as a graph?

Thats what I said in #1. If theyre looking for integer solutions then its unique and (3,21) but if its over the reals (well suitable domain for positive x, y) then you have a constraint/relationship. Can you post a pic of the original question and which section of the textbook does it come from as it seems a bit "hard" for a normal question.
(edited 2 months ago)
Reply 4
Original post by mqb2766
Thats what I said in #1. If theyre looking for integer solutions then its unique and (3,21) but if its over the reals (well suitable domain for positive x, y) then you have a constraint/relationship. Can you post a pic of the original question and which section of the textbook does it come from as it seems a bit "hard" for a normal question.

Verbatim the question is: "Given that (5√7 - √x)/(√7 - √x) = 8 + √y, where x any (sic?) y are positive integers and √y is a surd in its simplest form, find the values of x and y". The language of the question might have confused me as I am not sure if 'any' is a typo or not.
It's in the AQA Cambridge University Press Year 1 book on page 22, exercise 2B.
Apologies if I've had a massive oversight on this question.
Reply 5
Original post by Sorbus25
Verbatim the question is: "Given that (5√7 - √x)/(√7 - √x) = 8 + √y, where x any (sic?) y are positive integers and √y is a surd in its simplest form, find the values of x and y". The language of the question might have confused me as I am not sure if 'any' is a typo or not.
It's in the AQA Cambridge University Press Year 1 book on page 22, exercise 2B.
Apologies if I've had a massive oversight on this question.

Trying a few things it seems to be to be on the hard side. Unless Im missing something, dividing out you get
sqrt(y) = -7 + 4sqrt(7)/(sqrt(7)-sqrt(x))
and you can easily argue that x must be 2,3,4,5 or 6. Similarly you could take the 8 over and combine and get
sqrt(y) = (7sqrt(x)-3sqrt(7))/(sqrt(7}-sqrt(x))
and its relatively easy to see that 3 will work as the numerator will have a (sqrt(7)-sqrt(3)) factor so y will be an integer.

Neither are particularly a level methods though.
(edited 2 months ago)
Reply 6
Original post by Sorbus25
Verbatim the question is: "Given that (5√7 - √x)/(√7 - √x) = 8 + √y, where x any (sic?) y are positive integers and √y is a surd in its simplest form, find the values of x and y". The language of the question might have confused me as I am not sure if 'any' is a typo or not.
It's in the AQA Cambridge University Press Year 1 book on page 22, exercise 2B.
Apologies if I've had a massive oversight on this question.

Sleeping on it, it was the last thing in #1 to go with so rationalize to get
(35-x)/(7-x) + 4sqrt(7)sqrt(x)/(7-x)
As y is a surd in its simplest form, then equating the 8 to the first term gives ... and y is given by the second term.
Reply 7
Original post by mqb2766
Sleeping on it, it was the last thing in #1 to go with so rationalize to get
(35-x)/(7-x) + 4sqrt(7)sqrt(x)/(7-x)
As y is a surd in its simplest form, then equating the 8 to the first term gives ... and y is given by the second term.

Thank you! A lot more simpler than I realised. But a good lesson to not overcomplicate any questions

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