Source: AMC 10

a + a^-1 = 4

Find a^4 + a^-4


We really need LaTeX.

You square both sides of x+x^-1=4. You get x^2+2+x^-2=16. Then that means x^2+x^-2=14. Square both sides again. x^4+2+x^-4=196. Therefore, x^4+x^-4=194.

The answer is 194!

Correct!

How about this one?

x + y = 2

x^2 + y^2 = 3

Find x^3 + y^3.

(Yes, this is from Problems of the Day.)

86? Because if you divide the first equation into the second, you get x-y=3/2, and you can solve x=7/4 and y=1/4. And then you cube them, and add.

Sorry, that's incorrect.

edit: will post solution later.. just realized where this problem was from

I completely failed. The answer is 43/8. I divided wrong.

?? I got a way different answer.

Sorry, not 48/3. It is 5. Anyone have a solution, or should I post it?

x+y=2
x^2+y^2=3

cube x+y=2 to get x^3+y^3+3x^2y+3xy^2=8.
square x+y=2 to get x^2+2xy+y^2=4. Subracting x^2+y^2=3 from this, you find xy to be 1/2.
Now look at 3x^2y+3xy^2. This can be factored into 3xy(x+y), Since x+y=2, and xy=1/2, 3xy(x+y)=3. Subract this from x^3+y^3+3x^2y+3xy^2=8 to get x^3+y^3=5.

Correct! More on solving these in my blog.

For those who don't know about isabella's blog, the URL is:

http://www.artofproblemsolving.com/Forum/weblog.php?w=1172