for what values of m does the graph of y = mx² - 5x - 2 have no x - intercepts?\n○ m > -\\frac{25}{8}\n○ m <…

Explanation:

Step1: Recall the discriminant formula

For a quadratic equation $ax^{2}+bx + c = 0$, the discriminant $\Delta=b^{2}-4ac$. In the equation $mx^{2}-5x - 2=0$, we have $a = m$, $b=-5$, and $c=-2$.

Step2: Set up the condition for no x - intercepts

A quadratic function $y = ax^{2}+bx + c$ has no x - intercepts when $\Delta<0$. So, $\Delta=(-5)^{2}-4m(-2)<0$.

Step3: Simplify the inequality

First, calculate $(-5)^{2}-4m(-2)$: $25 + 8m<0$. Then, solve for $m$: $8m<-25$, so $m<-\frac{25}{8}$.

Answer:

$m<-\frac{25}{8}$