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: Recall discriminant for parabola

For a quadratic equation \(y = ax^2 + bx + c\), the discriminant is \(\Delta = b^2 - 4ac\). If the graph has no \(x\)-intercepts, \(\Delta < 0\). Here, \(a = m\), \(b = -5\), \(c = -2\).

Step2: Calculate discriminant

\(\Delta = (-5)^2 - 4(m)(-2) = 25 + 8m\)

Step3: Set discriminant less than zero

We want \(\Delta < 0\), so \(25 + 8m < 0\)

Step4: Solve the inequality

Subtract 25 from both sides: \(8m < -25\)
Divide by 8: \(m < -\frac{25}{8}\)

Answer:

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