To solve this problem, let’s analyze the quadratic equation:

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

We need to determine the values of $k$ for which this equation has exactly one root in the interval $(0, 1)$.

Step 1: Conditions for a Root in the Interval $(0, 1)$

For a quadratic equation $ax^2 + bx + c = 0$ to have exactly one root in an interval, there are a few possibilities:

1. The quadratic must have exactly one root, meaning the discriminant ($\Delta$) of the quadratic equation should be zero.
2. Alternatively, it should have two roots, one inside $(0, 1)$ and the other outside this interval.

In this case, the discriminant $\Delta$ of the equation must be non-negative for real roots: $\Delta = b^2 - 4ac$ where:

• $a = k^2 + 2$
• $b = -5k$
• $c = 4$

Thus, $\Delta = (-5k)^2 - 4(k^2 + 2)(4)$ $= 25k^2 - 16(k^2 + 2)$ $= 25k^2 - 16k^2 - 32$ $= 9k^2 - 32$

For real roots, $\Delta \geq 0$: $9k^2 - 32 \geq 0$ $9k^2 \geq 32$ $k^2 \geq \frac{32}{9}$ $k \geq \frac{\sqrt{32}}{3} \text{ or } k \leq -\frac{\sqrt{32}}{3}$

Evaluating the options with this result, we find that option 2: $3 \leq k \leq 4$ satisfies the required range for $k$.

Answer

The correct answer is: Option 2: $3 \leq k \leq 4$

Would you like a deeper explanation on any part of the process, or do you have additional questions?

Related Questions:

1. How do we determine if a quadratic equation has only one root?
2. What are the conditions for the discriminant of a quadratic equation to yield real roots?
3. Why do we check the discriminant when finding the nature of the roots?
4. How can we verify if both roots lie within a specified interval?
5. Can a quadratic equation have only one real root without the discriminant being zero?

Tip:

For quadratic equations, analyzing the discriminant is essential to understand the nature of the roots (real, complex, or identical).