( k − 3 ) x + 9

To solve the quadratic equation $x^2 + (k - 3)x + 9 = 0$ with roots $\alpha$ and $\beta$, we can use Vieta's formulas, which state:

• The sum of the roots: $\alpha + \beta = -(k - 3)$
• The product of the roots: $\alpha \beta = 9$

Given that $\alpha$ and $\beta$ are distinct and lie in the interval $(-6, 1)$, we start with the product:

$\alpha \beta = 9$

Since both roots are negative (as they are between $-6$ and $1$), we know that both roots must be less than $0$. This implies that $\alpha, \beta < 0$.

From the inequality $\alpha + \beta = -(k - 3)$, we have:

$\alpha + \beta < 1 + 1 = 2$

Thus,

$-(k - 3) < 2 \implies k - 3 > -2 \implies k > 1$

Next, we know:

$\alpha + \beta = -(k - 3) \implies k - 3 = -(\alpha + \beta)$

To find $k$ such that the roots are distinct and $\alpha, \beta \in (-6, 1)$, we analyze the discriminant of the quadratic equation:

$D = (k - 3)^2 - 4 \cdot 1 \cdot 9 = (k - 3)^2 - 36$

For the roots to be distinct, $D$ must be greater than $0$:

$(k - 3)^2 - 36 > 0$

This simplifies to:

$(k - 3)^2 > 36$

Taking the square root of both sides gives:

$|k - 3| > 6$

This results in two cases:

1. $k - 3 > 6 \implies k > 9$
2. $k - 3 < -6 \implies k < -3$

Since we already established $k > 1$, we only consider the first case:

$k > 9$

Now, let's also check that the roots can exist in the interval $(-6, 1)$ given $k > 9$.

To confirm that this holds, let's verify if the limits hold:

• For $k = 10$ (just above $9$):
  • The quadratic becomes $x^2 + 7x + 9 = 0$.
  • The discriminant is $7^2 - 36 = 49 - 36 = 13$, which is positive.

Calculating the roots:

$x = \frac{-7 \pm \sqrt{13}}{2}$

Calculating the approximate values:

$\alpha, \beta = \frac{-7 \pm 3.605}{2}$ $\alpha \approx -1.697, \quad \beta \approx -3.303$

Both roots are indeed in the interval $(-6, 1)$.

Thus, the value of $k$ is:

$\boxed{10}$

Further Questions:

1. How does the discriminant affect the nature of the roots?
2. What are the implications of the roots being distinct?
3. How can we find the specific values of $\alpha$ and $\beta$ for other $k$ values?
4. Can you derive the roots for different values of $k$ within the specified range?
5. What are the graphical representations of the quadratic for different $k$?

Tip: When solving quadratic equations, always check the conditions for real and distinct roots through the discriminant.