I notice that both sides of the equation have the same base, 3, raised to different powers. That means that the powers must be equal for the equation to hold. So I can set $2x^{2}-5x+2 = 2x^{2}+7x-4$ and solve for $x$. To do that, I subtract $2x^{2}$ from both sides and get $-5x+2 = 7x-4$. Then I add $5x$ to both sides and get $2 = 12x-4$. Next, I add $4$ to both sides and get $6 = 12x$. Finally, I divide both sides by $12$ and get $x = \frac{6}{12} = \frac{1}{2}$. 
For what value of $x$ does $3^{2x^{2}-5x+2} = 3^{2x^{2}+7x-4}$? Express your answer as a common fraction.