Question 644404
a³+b³=72, a²-b²=12
 find the value of a and b
<pre>

(1)     a³+b³ = 72
(2)     a²-b² = 12

From (2)   b² = a²-12
(3)         b = ±<font face="symbol">Ö</font><span style="text-decoration: overline">a²-12</span>

Substituting the negative solution in (1) and simplifying
gives a 6th degree equation with only imaginary solutions.
I will not deal with the imaginary solutions. So we will
only consider

(3)         b = <font face="symbol">Ö</font><span style="text-decoration: overline">a²-12</span>

Substituting this positive square root and simplifying gives
this: 

   a<sup>4</sup> - 4a<sup>3</sup> - 12a<sup>2</sup> + 192 = 0

which factors as

 (a-4)(a<sup>3</sup>-12a-48) = 0

This has rational solution a=4.  It also has one irrational
real solution and two imaginary solutions.  Substituting a=4
this in (3) gives b=2

Real rational solution:  a=4, b=2

I will not deal with the irrational real solution or the imaginary 
solutions.

Edwin</pre>