To determine for what values of $m$ the graph of the quadratic equation $y = mx^2 - 5x - 2$ has no x-intercepts, we need to analyze the discriminant of the quadratic equation.

The quadratic equation is of the form $y = ax^2 + bx + c$, where:

• $a = m$
• $b = -5$
• $c = -2$

The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = b^2 - 4ac$

In our case: $\Delta = (-5)^2 - 4(m)(-2)$ $\Delta = 25 + 8m$

For the quadratic to have no x-intercepts, the discriminant must be less than zero: $25 + 8m < 0$

Solving for $m$: $8m < -25$ $m < -\frac{25}{8}$

Thus, the graph of $y = mx^2 - 5x - 2$ will have no x-intercepts for $m < -\frac{25}{8}$.

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Would you like more details on any part of this solution or additional clarification?

Here are five related questions you may find helpful:

1. What happens when $m = -\frac{25}{8}$?
2. How does the discriminant relate to the number of x-intercepts?
3. For what values of $m$ does the quadratic have exactly one x-intercept?
4. Can you graph the equation for a specific value of $m$ to visualize it?
5. How does the value of $m$ affect the shape of the parabola?

Tip: If the discriminant is zero, the quadratic will have exactly one real solution, meaning the parabola touches the x-axis at one point.