a^6-a^2=a^4-1

Question

trying to find the variable

Answer 1

a⁶-a²=a⁴-1 can be written a⁶-a⁴-a²+1=0.

Let x=a², then x³-x²-x+1=0. When x=1, 1-1-1+1=0 so x=1 is a solution.

Using synthetic division we can reduce the cubic to a quadratic:

1 | 1 -1 -1    1

     1  1  0 | -1

     1  0 -1 | 0 = x²-1=(x-1)(x+1), so x³-x²-x+1=(x-1)²(x+1).

x-1=0, x=1 means that a²=1, so a=-1 or 1. x+1=0, x=-1=a² only has complex solutions (-i and i).

CHECK REAL SOLUTIONS

a=1 or -1: 1-1=1-1✔️