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


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.

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

Explore how to find the values of m for which the graph of the quadratic equation y = mx^2 - 5x - 2 has no x-intercepts. This problem involves understanding the discriminant and its role in determining the nature of quadratic solutions. Suitable for students in Grades 9-11.

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