The sum of all real values of x satisfying (x^2-5x+5)^x^2+4x-60=1 is:

Question

The sum of all real values of $x$ satisfying $(x^2-5x+5)^{x^2+4x-60}=1$ is:

Accepted Answer

Case I: base $=1$: $x^2-5x+5=1 \implies x=1,4$.

Case II: base $=-1$ and exponent even: $x^2-5x+5=-1 \implies x=2,3$; exponent at $x=3$ is odd, rejected; $x=2$ valid.

Case III: exponent $=0$, base $
e0$: $x^2+4x-60=0 \implies x=-10,6$.

Sum of all valid $x$: $1+4+2-10+6=3.$

Solutions · 1