How do you solve for x in this quadratic equation? : 5x^2 – 34x + 24 = 0

Quadratic formula
factoring
graphing 
completing the square
factoring by grouping
Rational Roots Theorem
synthetic division

Take a look at 5x^2 – 34x + 24 = 0.  The last term could have been the result of these different possible muliplications:  1*24, 2*12, 3*8, 4*6.  The leading term is 5, whose factors are 5 and 1.  Thus, possible rational roots would be
4/5 (the 4 is a factor of 24 and the 5 is a factor of 5) and 6/1 (the 6 is a factor of 24 and the 1 is a factor of 5).  

Using synth. div. to check whether 6 is actually a root:
     
6  /    5      -34          24
                  30         -24
   
         5       -4             0

since the remainder is 0, we can safely call 6 a "root."  
Note the remaining coefficients, 5 and -4:  

They correspond to the factor 5x - 4.  If we set this difference = to 0, and solve for x, we get  x = 4/5 (which is correct).

The roots of 5x^2 – 34x + 24 = 0  are  x = 4/5 and x= 6/1.