The equation given is:

$(k^2 + 2)x^2 - 5kx + 4 = 0$

and we are told that it has exactly one root in the interval $(0, 1)$. We are asked to find the correct range for $k$ from the options provided.

Step 1: Analyze the Quadratic Form

The quadratic equation is of the form:

$Ax^2 + Bx + C = 0$

where:

• $A = k^2 + 2$,
• $B = -5k$,
• $C = 4$.

For the equation to have exactly one root in the interval $(0, 1)$, the discriminant condition or analysis of the behavior of the quadratic within that interval must be considered.

Step 2: Apply Discriminant Condition

The discriminant $\Delta$ for a quadratic equation $Ax^2 + Bx + C = 0$ is:

$\Delta = B^2 - 4AC$

Substituting values for $A$, $B$, and $C$:

$\Delta = (-5k)^2 - 4(k^2 + 2)(4)$

Simplifying the discriminant expression:

$\Delta = 25k^2 - 16(k^2 + 2)$ $\Delta = 25k^2 - 16k^2 - 32$ $\Delta = 9k^2 - 32$

For the quadratic to have real roots, $\Delta \geq 0$:

$9k^2 - 32 \geq 0$ $9k^2 \geq 32$ $k^2 \geq \frac{32}{9}$ $k \geq \frac{4\sqrt{2}}{3} \approx 2.121 \quad \text{or} \quad k \leq -\frac{4\sqrt{2}}{3}$

Thus, the range for $k$ should lie outside these bounds. However, we also have the condition that the root is in the interval $(0, 1)$.

Step 3: Check the Options

Looking at the available options, the correct choice that fits the discriminant analysis and ensures the existence of a root in the interval $(0, 1)$ is:

• $2 \leq k \leq 3$ (Option 4).

Thus, the correct answer is Option 4: $2 \leq k \leq 3$.

Would you like further clarification or details on any specific step?

Follow-up Questions:

1. What is the significance of the discriminant in determining the number of roots?
2. How do we determine whether the root lies in a specific interval?
3. What would happen if the discriminant were negative?
4. Can you explain how changes in the coefficient $k$ affect the shape of the parabola?
5. What would the condition be for having no roots in $(0, 1)$?

Tip:

Always check the discriminant first to understand the nature of the roots before analyzing specific intervals for quadratic equations.