Math Geniuses!! Please help me with this Problem of the Week...?

Question

Math Geniuses!! Please help me with this Problem of the Week...?
I'd be passing Trig if i get this one! So please help me out as much as you can, and i'll love you forever :)
WARNING: THIS IS SUPER HARD. NO ONE CAN FIGURE IT OUT IN MY CLASS (and i have a smart class)

Suppose that X and Y are two real numbers such that X - Y=2 and X^2 + Y^2=8. Find X^3 - Y^3

Thanks guys!!! Even if you dont get it, i give you credit for trying!!!

Answer 1

Luken is right.


The answer is 20.

Answer 2

I don't know how much help this will be, but it is worth noting that x^3 - y^3 can be factored as shown below.

X^3 – Y^3 = (X – Y)(X^2 + XY + Y^2)

We can rearrange this as

X^3 – Y^3 = (X – Y)([X^2 + Y^2] + XY)

From the information given we know that X - Y = 2
and X^2 + Y^2 = 8, so all we need is to find XY. Let's use
X - Y = 2, and square both sides.

(X - Y)^2 = 2^2
X^2 -XY -XY + Y^2 = 4, do some strategic grouping
[X^2 + Y^2] - 2XY = 4, from above we know that X^2 + Y^2 = 8
8 - 2XY = 4, subtract 8 from both sides
8 - 8 - 2XY = 4 - 8, do the math
-2XY = -4, divide both sides by -2
-2XY/-2 = -4/-2, do the math
XY = 2

Now using the rearranged equation from above we can do some substitution

X^3 – Y^3 = (X – Y)([X^2 + Y^2] + XY)
= (2)(8 + 2)
= (2)(10)
= 20

Always remember that for these type of problems you will most likely have to do some algebraic manipulation.

Answer 3

First, you'll need this little identity:
(X – Y)² = X² - 2XY + Y²
-2XY = (X – Y)² - (X² + Y²)
-XY = [(X – Y)² - (X² + Y²)]/2

Now, multiply the two by FOIL method
(X – Y) ( X² + Y² ) = X³ + XY² - X²Y – Y³

Rearrange:
X³ – Y³ = (X – Y) ( X² + Y² ) + X²Y - XY²
X³ – Y³ = (X – Y) ( X² + Y² ) – XY(X – Y)

Now replace -XY with the identity from above:
X³ – Y³ = (X – Y) ( X² + Y² ) – [(X – Y)² - (X² + Y²)]/2 ( X-Y)

Substitute in the values.
X³ – Y³ = (X – Y) ( X² + Y² ) – [(X – Y)² - (X² + Y²)]/2 ( X-Y)
X³ – Y³ = (2) (8) – [(2)² - (8)]/2 (2)
X³ – Y³ = 16 – [4 - 8]
X³ – Y³ = 20

For fun, try to solve for X and Y. I haven't tried yet.

Answer 4

Well, it isn't as super hard as you all think.
Look, let's factor x^3 - y^3= (x-y)(x^2 + xy + y^2).
Now we know x-y and x^2 + y^2, so all we need
to find is xy.
To do this note that (x-y)^2 = x^2 -2xy + y^2 = 4,
x^2 + y^2 = 8.
Subtracting, we get
-2xy = -4 or xy = 2
So x^3 - y^3 = 2(8 + 2) = 20.
Hope that helps!

Answer 5

I'm just going to start you off... hopefully in the right direction

X - Y = 2
X = 2 + Y

X^2 + Y^2 = 8
Substitute (2+Y) for X you get
(2+Y)^2 + Y^2 = 8
4 + 4Y + Y^2 + Y^2 = 8
2Y^2 + 4Y + 4 = 8
2Y^2 + 4Y - 4 = 0
2(Y^2 + 2Y - 2) = 0

Answer 6

i think that the "^" means to the power... maybe?
so are you saying:
if X - Y = 2
then X^3 + Y^3 = 12

maybe.... i don't really understand what your saying... but then again i haven't taken Trig yet.

Answer 7

we know that x^2+y^2-2xy=(x-y)^2
=>8-2xy=(2)^2=4
=>-2xy=4-8=-4
->xy=2
Now x^3-y^3=(x-y)(X^2+y^2+xy)
=(2)(8+2)
=2X10=20

Answer 8

I'll help you but what is it equal to?

Answer 9

wait what does the little thingy mean? " ^ "?

Answer 10

what are these supposed to mean : ^ ? is it multiplying?