solve a^2-b^2, while a^3-b^3=513 and ab=54.?

Answer 1

a^3 - b^3 = (a-b) (a^2 + ab + b^2)
Assuming positive integer solutions, we want two numbers that multiply to 54, whose difference is a factor of 513.
513 = 3^3.19
54 = 2.3^3
Factor combinations of 54 are:
1, 54 - difference 53 doesn't work
2, 27 - difference 25 doesn't work
3, 18 - difference 15 doesn't work
6, 9 - difference 3 is a factor of 513.

So we must have a = 9 and b = 6 for this to be possible. Check: 9^3 - 6^3 = 729 - 216 = 513. So these are the correct figures. So a^2 - b^2 = 81 - 36 = 45.

Elder God: nice general solution. I agree there SHOULD be a way to use difference of cubes, but I don't really see it.

Answer 2

a^2 - b^2 = 45 is just one possibility.

Here's why.
Start with ab = 45. Solve for either a or b. (it doesn't matter which one, the result will be the same.) Also don't bother to cube the 54, it just becomes a big ugly number (157,464) that will be harder to factor.

a = 54/b (Put this into your cubed equation)
a^3 - b^3 = 513 then becomes
(54/b)^3 - b^3 = 513
(54)^3/b^3 - b^3 = 513 Next clear denominators by multiplying by b^3
54^3 - b^6 = 513b^3 Set equal to zero.
b^6 - 513b^3 - 54^3 = 0 (this is a quadratic look-alike)
Let u = b^3
u^2 - 513u - 54^3 = 0 (see - a quadratic equation!!)
(u - 9^3)(u + 6^3) = 0
u = 9^3 OR u = -6^3 (go back to "b")
b^3 = 9^3 OR b^3 = -6^3
b = 9 OR b = -6

Since a = 54/b = 54/9 = 6
OR a = 54/b = 54/(-6) = -9

Therefore, a^2 - b^2 = 6^2 - 9^2 = 36 - 81 = -45
OR a^2 - b^2 = (-9)^2 - (-6)^2 = 81 - 36 = 45

I hope that helps.

Answer 3

There should be a nice way using difference of cubes, but I'm just going to use brute force.

a^3-b^3 = 513 (equation1)
ab = 54 (equation2)

So from the 2nd equation, a = (54/b)

Substitute that into the first equation:
(54/b)^3 - b^3 = 513
Simplify:
157464 - b^6 = 513b^3
so, b^6 + 513b^3 - 157464 = 0
Let x=b^3
Our equation becomes:
x^2 + 513x - 157464 = 0
Factorize:
(x-216)(x+729)=0
So either: (x-216)=0 OR (x+729)=0
Solve for x: x=216,-729

But x=b^3
so b^3 = 216,-729
Take the cube root:
b = 6, -9

Substitute these values for b into equation2:
(i) ab = 54
a*6 = 54
a = 54/6 = 9

(ii) ab = 54
a*-9 = 54
a = -6

So our solutions are: (a,b) = (9,6) or (a,b) = (-6,-9)
Then we have 2 solutions:
(i) a^2 - b^2 = 9^2 - 6^2 = 45
(ii) a^2 - b^2 = (-6)^2 - (-9)^2 = -45

Answer 4

I'm not trying to be trivial - but I'm w/out aids... except: 6 X 9 = 54. Start there. a = 9, .....

Answer 5

a^2 - b^2 = 45