Question

For what values of m does the graph of y = mx2 – 5x – 2 have no x-intercepts?

Answer 1

Take a look at   y = mx^2 – 5x – 2.   This is a quadratic equation with a=m, b= -5 and c= -2.  What is the discriminant, b^2 - 4ac?  (-5)^2 - 4(m)(-2).

There will be 2 complex roots if this discrim. is < 0.  Thus, solve 

25 + 8m < 0:   

Subtr. 25 from both sides:              8m < -25
Div. both sides by 8:                         m < -25/8

Conclusion:  If m is less than -25/8, there will be no real roots (and thus, no x-intercepts).