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Need Help with Surds Questions

Given that the point with coordinates (1+sqrt3 , 5sqrt3) lies on the curve y=2x^2+px+q
Find the values of the rational constants p and q

And also this one, simplify:
(sqrt p+1 - sqrt p)(p+0.5+sqrt p^2+p) all over sqrt p+1 +sqrt p

Answers are: p=1 q=9 and the simplify is 1/2
I’m aware these are fairly simple questions, but we’ve moved on from the surds topic now and everything else was fairly comprehendable but I am unsure how to answer these 2 after doing the first simple parts.

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Reply 1

mqb2766

Original post by Jason03

What do you get when you sub the point into the first equation?

Reply 2

Jason03

Original post by mqb2766

What do you get when you sub the point into the first equation?

5sqrt3 = 8+4sqrt3+p+psqrt3 +q

Reply 3

mqb2766

Original post by Jason03

5sqrt3 = 8+4sqrt3+p+psqrt3 +q

So choose p & q (rational) to make the surds and rational parts match on each sude.

Reply 4

Jason03

Original post by mqb2766

So choose p & q (rational) to make the surds and rational parts match on each sude.

Not too sure what you mean, just answer it by trial and error?

Reply 5

mqb2766

Original post by Jason03

Not too sure what you mean, just answer it by trial and error?

No, for the surds, what multiplies sqrt(3) on each side of the equation. They must balance.

Reply 6

Jason03

Original post by mqb2766

No, for the surds, what multiplies sqrt(3) on each side of the equation. They must balance.

Yeah so it has to be p=1 and therefore q=9 but there isn’t any like law or anything I need to know, just that the 5sqrt3=4sqrt3+psqrt3 and then 0=8+p+q it’s that simple?

Reply 7

mqb2766

Original post by Jason03

Yeah so it has to be p=1 and therefore q=9 but there isn’t any like law or anything I need to know, just that the 5sqrt3=4sqrt3+psqrt3 and then 0=8+p+q it’s that simple?

Yes it's that simple. It just used basic arithmetic laws on rational and irrational (surds) numbers.

Can you upload a pic of the second question, the layout is confusing.

(edited 3 years ago)

Reply 8

Jason03

Original post by mqb2766

Yes it's that simple. It just used basic arithmetic laws on rational and irrational (surds) numbers.

Can you upload a pic of the second question, the layout is confusing.

(edited 3 years ago)

Reply 9

mqb2766

Original post by Jason03

What do you think you can do to rationalise it?
Look at the first term in the numerator and the complete denominator.

Reply 10

Jason03

Original post by mqb2766

What do you think you can do to rationalise it?
Look at the first term in the numerator and the complete denominator.

I rationalised and got to 1(p+1/2+sqrtp^2+p)
then to 2p+1/2+sqrt p

Reply 11

mqb2766

Original post by Jason03

I rationalised and got to 1(p+1/2+sqrtp^2+p)
then to 2p+1/2+sqrt p

Upload your working?

Reply 12

mqb2766

Original post by Jason03

I rationalised and got to 1(p+1/2+sqrtp^2+p)
then to 2p+1/2+sqrt p

It s also with thinking how you can represent the second term in the numerator in terms of sqrt(p) terms. That's probably the most elegant way to solve the problem.

(edited 3 years ago)

Reply 13

Jason03

Original post by mqb2766

It s also with thinking how you can represent the second term in the numerator in terms of sqrt(p) terms. That's probably the most elegant way to solve the problem.

I’ve obviously done something wrong as the surd shouldn’t be negative and also it doesn’t look like it would cancel down to 1/2

Reply 14

mqb2766

For the 2nd term on numerator
((p + 1/2 + sqrt(p^2+p))
1/2(2p + 1 + 2sqrt(p)sqrt(p+1))
1/2(p+1 + 2sqrt(p)sqrt(p+1) + p)
1/2(sqrt(p+1) + sqrt(p))^2

Easy to finish?

Reply 15

Jason03

Original post by mqb2766

For the 2nd term on numerator
((p + 1/2 + sqrt(p^2+p))
1/2(2p + 1 + 2sqrt(p)sqrt(p+1))
1/2(p+1 + 2sqrt(p)sqrt(p+1) + p)
1/2(sqrt(p+1) + sqrt(p))^2

Easy to finish?

Not sure how sqrt(p^2+p) becomes sqrt(p)sqrt(p+1) can you explain? Sorry for being a pest