That's it

Algebra Level 4

$\large {a^3 = 3(a^2 +b^2) - 25\\ b^3 = 3(b^2 +c^2) - 25\\ c^3 = 3(c^2+a^2) - 25}$

If $a, b, c$ are positive reals that satisfy the above system of equations, find $abc$ .

The answer is 125.

1 solution

Tamir Dror
Nov 28, 2015

We have to assume that a=b=c because the Equaltons are simmilar C^3=3c^2+3c^2-25 // C^2(c-6)=-25// c c (c-6)=5 5 -1

So c=5 because it repeats twice, And since a=b=c, abc is 5 5 5=125

Moderator note:

For a complete solution, we have to show that the value of $abc$ is independent of any other assumptions that we have made. In particular, we cannot assume that $a = b = c$ .

Please state the reason why a=b=c.

Raven Herd - 5 years, 6 months ago

As I said, the equaltons are simillar and you can obviously see they are switchable.

Tamir Dror - 5 years, 6 months ago

Try this That's it - 2

Raven Herd - 5 years, 6 months ago

@Raven Herd – I solved it, but I can't post there.

Tamir Dror - 5 years, 6 months ago

@Tamir Dror – Why not ?????

Raven Herd - 5 years, 6 months ago

@Raven Herd – Because the level of it is still pending :(

Tamir Dror - 5 years, 6 months ago

@Tamir Dror – You type it here.

Raven Herd - 5 years, 6 months ago

@Raven Herd – Alright. So we can make the first equalton this way: y(x-1)(x+1)+x=19 x cant be 1, because the equalton wont be right. So we have to find x which will make 19-x a number with 3 prime factors. Try x=3 because 16=4(4)(1) or 8(2)(1) or 4(2)(2). The only sulotion which fits is the last one. Put it in the equalton: 16=8y // y=2 // 4z+2+z=27 //z=5 //

Check it: 75+3-5=73 // 73=73//

So x+y+z is 10

Tamir Dror - 5 years, 6 months ago

That's it

The answer is 125.

1 solution

Moderator note:

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