Question

for what values of m does the graph of $y = mx^2 - 5x - 2$ have no x-intercepts?\
\\(\bigcirc\\ m> -\frac{25}{8}\\)\
\\(\bigcirc\\ m< -\frac{25}{8}\\)\
\\(\bigcirc\\ m< \frac{25}{8}\\)\
\\(\bigcirc\\ m> \frac{25}{8}\\)

Explanation:

Step1: Identify discriminant condition

A quadratic $y=ax^2+bx+c$ has no x-intercepts when its discriminant $\Delta = b^2-4ac < 0$, and $a
eq0$.

Step2: Substitute coefficients

Here, $a=m$, $b=-5$, $c=-2$.
$\Delta = (-5)^2 - 4(m)(-2) < 0$

Step3: Simplify the inequality

$25 + 8m < 0$

Step4: Solve for m

$8m < -25$
$\frac{8m}{8} < \frac{-25}{8}$
$m < -\frac{25}{8}$
Also, confirm $m
eq0$ is included in this range.

Answer:

$m < -\frac{25}{8}$ (corresponding to the second option)