Crazy indices

Algebra Level 4

Find the sum of all positive real $x$ that satisfies

$\sqrt{x}( 9^{\sqrt{x^{2}-3}} - 3^{\sqrt{x^{2}-3}}) = 3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2} - 3}+1} + 6\sqrt{x} - 18$ ,

1 solution

Ww Margera
Sep 24, 2014

Let $\sqrt(x)=y$ , $3^{x^2-3}=z$ . Simplifying the expression, $(y-3)(z-3)(z+2)=0$ . (z+2) will never be 0. y=3 gives x=9, z=3 gives x=2;