4unit poly question... Help please (1 Viewer)

[conics2008]

Hi.... Here is the question.

The roots of the Quadratic x^2 - 8x +20 = 0 are alpha and beta.

Form an equation whose roots are alpha + 1 / alpha , beta + 1 / beta.

................................................................................................................

Thanks in advanced =)

 

[foram]

Hi.... Here is the question.

The roots of the Quadratic x^2 - 8x +20 = 0 are alpha and beta.

Form an equation whose roots are alpha + 1 / alpha , beta + 1 / beta.

................................................................................................................

Thanks in advanced =)

I think this is how somebody showed me once in the 3U thread. I've never tried this so i don't know what i'm doing. I don't do 4U yet... so i dunno.

let x=a+1 /a
ax= a+1
ax-a=1
a(x-1) =1
a = 1/(x-1)

P(a)= 1/(x-1)^2 - 8/(x-1) + 20 = 0
1/(x^2 -2x +1) - (8x-8)/ (x^2 -2x +1) = -20
1-8x-8 = -20(x^2 - 3x +1)
-8x -7 = -20x^2 +60x -20
20x^2 -68x +14 = 0
10x^2 -34x + 7 = 0

EDIT: I remember, tommykins showed me how to do this. I think he called it Polynomial transformation?

 

[conics2008]

I think this is how somebody showed me once in the 3U thread. I've never tried this so i don't know what i'm doing. I don't do 4U yet... so i dunno.

let x=a+1 /a
ax= a+1
ax-a=1
a(x-1) =1
a = 1/(x-1)

P(a)= 1/(x-1)^2 - 8/(x-1) + 20 = 0
1/(x^2 -2x +1) - (8x-8)/ (x^2 -2x +1) = -20
1-8x-8 = -20(x^2 - 3x +1)
-8x -7 = -20x^2 +60x -20
20x^2 -68x +14 = 0

EDIT: I remember, tommykins showed me how to do this. I think he called it Polynomial transformation?

I know how to do it but look at you're working out its wrong


i know this part x = a + 1/a now you have to make a the subject how ? I dont knwo

this is what you did x = a + 1/a >> ax = a +1 ??????? i dont know how you came to that conculsion ???

its a fraction 1/a

x= a + 1/a >>>> xa = a^2 + 1 >> ???? I get stuck here...

 

[foram]

I know how to do it but look at you're working out its wrong


i know this part x = a + 1/a now you have to make a the subject how ? I dont knwo

this is what you did x = a + 1/a >> ax = a +1 ??????? i dont know how you came to that conculsion ???

its a fraction 1/a

x= a + 1/a >>>> xa = a^2 + 1 >> ???? I get stuck here...

oops, i thought you meant (a+1)/a.

 

[Sandchairs]

MASSIVE EDIT:
sure change teh question after i answer it...
btw
u made a mistake when u said
1-8x-8 = -20(x^2 - 3x +1) <whered the 3 come from???

 

[foram]

Just ignore the foolishness of a person who doesn't do 4U math yet. T_T

I wish I could do 4U in year 11.

 

[Sandchairs]

Just ignore the foolishness of a person who doesn't do 4U math yet. T_T

I wish I could do 4U in year 11.

uh no u dont...
though id have to say if nothing else its made my life more interesting cause i get a lift home with my teacher after the lessons...

 

[Slidey]

Hi.... Here is the question.

The roots of the Quadratic x^2 - 8x +20 = 0 are alpha and beta.

Form an equation whose roots are alpha + 1 / alpha , beta + 1 / beta.

................................................................................................................

Thanks in advanced =)

First of all, there aren't any real roots. They are complex. Not that that matters.

ab=20
a+b=8

If you actually meant a+1/a and b+1/b, then:
product: (a+1/a)(b+1/b)=ab+a/b+b/a+1/(ab)=ab+1/(ab)+(a^2+b^2)/(ab)
=ab+1/(ab)+[(a+b)^2-2ab]/(ab) = 20+1/20+(64-40)/20=20+25/20=85/4
sum: a+b+1/a+1/b=a+b+(a+b)/(ab) = 8+8/20 = 42/5

New polynomial: y=x^2-(42/5)x+85/4

Alternatively: use the substitution u=x+1/x --> ux=x^2+1, x=(-u+sqrt(u^2-4))/2
On second thought, don't bother with this method. (Foram made a mistake by not multiplying a by a, unless he meant roots (a+1)/a, and if that's what he was trying to solve, he should have used brackets.)

Confirmation though roots:
x^2-8x+20=0
2x=8+/-sqrt(64-80)
x=4+/-2i
sum of roots: 8
product of roots: 16+4=20

sum of 4-2i+1/(4-2i) + 4+2i+1/(4+2i) = 8 + 8/20=42/5
product of (4-2i+1/(4-2i))(4+2i+1/(4+2i)) = (20+[4-2i]/[4+2i]+[4+2i]/[4-2i]+1/20) = 20+1/20+(12+12)/20 = 85/4

Yay.

This is a 2unit question, BTW.

 

[Slidey]

Just ignore the foolishness of a person who doesn't do 4U math yet. T_T

I wish I could do 4U in year 11.

You did pretty well. You were just solving the wrong problem. Correct method, though.

 

[conics2008]

a huge edit lol.... my bad.. i miss read you're new equation... thanks alot.. and I got confused with you're working out lol.. I re-wrote it thanks alot slidey =) you're my HERO LoL..................


Heres $5.. dont spend it on candy!

 

[Slidey]

No problem! Good luck.

 

[kooltrainer]

2 unit question? sorry

 

[Slidey]

It involves no 4unit or 3unit concepts.

It only involves the product and sum of roots of a quadratic. Perhaps one could consider it a 3u question on the grounds it uses the identity a^2+b^2=(a+b)^2-2ab. *shrug*.

It's a hard question, granted, but... yeah.

 

[jcurry]

wouldnt it be considered a harder 2U question and therefore would only be applied in a 3U test, i honestly can not see 95% of 2U people who would be able to do it

 

[Slidey]

Perhaps, but 2unit students should definitely be solving questions like that as preparation for their 2unit tests. It's the perfect question because it elegantly brings together various different components of your course into one problem. While you shouldn't be expected to know how to solve it by yourself at a 2u level, you should both attempt to solve it, and know howsolve it fully yourself (even if you were given the solution earlier).

 

[jcurry]

i see what you mean but i reckon 90% of people wouldnt attempt to learn to do stuff above and beyond the difficulty they need to do in the HSC
well at my school anyway

 

[Slidey]

Their loss.