Need Help with Surds Questions

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#1
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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4 weeks ago
#2
(Original post by Jason03)
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.
What do you get when you sub the point into the first equation?
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#3
(Original post by mqb2766)
What do you get when you sub the point into the first equation?
5sqrt3 = 8+4sqrt3+p+psqrt3 +q
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4 weeks ago
#4
(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.
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#5
(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?
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4 weeks ago
#6
(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.
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#7
(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?
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4 weeks ago
#8
(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.
Last edited by mqb2766; 4 weeks ago
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#9
(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.
Last edited by Jason03; 4 weeks ago
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4 weeks ago
#10
(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.
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#11
(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
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4 weeks ago
#12
(Original post by Jason03)
I rationalised and got to 1(p+1/2+sqrtp^2+p)
then to 2p+1/2+sqrt p
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4 weeks ago
#13
(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.
Last edited by mqb2766; 4 weeks ago
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#14
(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
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4 weeks ago
#15
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?
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#16
(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
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4 weeks ago
#17
(Original post by Jason03)
Not sure how sqrt(p^2+p) becomes sqrt(p)sqrt(p+1) can you explain? Sorry for being a pest law
laws of indices:
sqrt(ab) = sqrt(a)sqrt(b)
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#18
(Original post by mqb2766)
laws of indices:
sqrt(ab) = sqrt(a)sqrt(b)
Yeah but surely wouldn’t it go to sqrt(p^2)sqrt(p) which would just become (p)sqrt(p) or am I missing something
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4 weeks ago
#19
(Original post by Jason03)
Yeah but surely wouldn’t it go to sqrt(p^2)sqrt(p) which would just become (p)sqrt(p) or am I missing something
where have you got a sqrt(p^2) from?
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4 weeks ago
#20
(Original post by Jason03)
Yeah but surely wouldn’t it go to sqrt(p^2)sqrt(p) which would just become (p)sqrt(p) or am I missing something
p^2+p = p(p+1)
Then root.
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