Determine all values of a that will cause the parabola y=ax^2+10x+1 to cross the x axis at two distinct locations.

What you must do for this case is find the cut points with the x axis.
 For this, we evaluate when:
 y = 0
 0 = ax ^ 2 + 10x + 1
 We solve the polynomial using the resolver:
 x = (- b +/- root (b ^ 2 - 4ac)) / 2a
 We see where the root is defined:
 b ^ 2 - 4ac> = 0
 10 ^ 2 - 4 * a * 1> = 0
 100> = 4 * a * 1
 100/4> = a
 25> = a
 We go where the denominator is not defined:
 2a = 0
 a = 0
 Therefore the values of a must be between 0 and 25, written mathematically:
 a = (0, 25]
 
 a = (0, 25]