Answer:
There are four possible numbers:
-12.9 and 16.22
12.92 and 16.22
-12.92 and 16.22
12/29 and -16.22
Explanation:
To be honest, this should be easily solved using the two equations that can be formed from the problem statement. Let x and y be the two numbers.
"The difference between the squares of two numbers is 96" can be expressed as y^2-x^2=96.
"Twice the square of the second number subtracted from the square of the first number is 71." may be written as 2x^2-y^2 = 71.
By rearranging to isolate x or y in either of the equations, that definition of x or y may be substituted in the first equation and then solved for the remaining variable. But I would up needing the square root of a negative number. Adding "i" is OK, but the derivation wasn't as easy as I had expected. So I decided to graph the two equations and find the points at which the two lines intersect.
See the attached resulting graph. The two equations formed parabolas that open in both directions (up or down). This results in 4 intersection points, all of which could be solutions. I read the points as best I could and entered then in the table on the attachment. The spreadsheet excerpt also shows that the points all satisfy the two equations.
The solution may be found mathematically, and I hope someone adds that approach to the answers. Someone with more patience than I posess.