#### Question

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

Question

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

The x-intercepts of a parabola occur when y = mx² – 5x – 2 equals zero. Therefore, we need to solve the equation:

mx² – 5x – 2 = 0

For the graph of the equation to have no x-intercepts, the discriminant "b² - 4ac" must be less than zero (since a and b are both non-zero in this equation). The discriminant of this quadratic equation is:

b² - 4ac = (-5)² - 4(m)(-2) = 25 + 8m

For the equation to have no x-intercepts, we need:

25 + 8m < 0

Subtracting 25 from both sides of the inequality gives:

8m < -25

Dividing both sides by 8 (remembering to flip the inequality since we are dividing by a negative number) gives:

m > -25/8

Therefore, for values of m greater than -25/8, the graph of y = mx² – 5x – 2 has no x-intercepts.