Q:

How do I find the solution set to the inequality $3x^2-4x+5<0$

How do I find the solution set to the inequality $3x^2-4x+5\lt 0$?
I know the basic rules for finding the solution set, but I'm not sure what step I should do first to solve the problem, like if I should subtract 5 from both sides first or go straight to factoring. 

A:

Note that the discriminant is negative. So the solutions are complex. That means the graph of the function $f (x)=3x^2-4x+5$ never touches the  $x $-axis. Also since the coefficient of $x^2$ is $3$,a positive integer, the function itself is always positive. (Think of  $x$ tending to large numbers and the domination of  $x^2$ over  $x$.)

A:

One way is to observe that
$$3x^2-4x+5 < 0 \iff 9x^2-12x+15 < 0.$$
But
$$9x^2-12x+15 = (3x-2)^2 + 11 \ge 11.$$
Therefore, ....

