If 
α
,
β
 are the roots of the quadratic equation 
x
2
+

(
k
−
3
)
x
+
9
=
0
 
(
α
≠
β
)
. If 
α
,
β
∈
(
−
6
,
1
)
 then find the value of 
k
.

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.

If α, β are the roots of the quadratic equation x^2 + (k - 3)x + 9 = 0 (α ≠ β). If α, β ∈ (-6, 1) then find the value of k.

Learn how to determine the value of k for a quadratic equation with roots in the interval (-6, 1). This problem involves using Vieta's formulas and the discriminant to analyze the conditions for real, distinct roots. Ideal for students in Grade 10-12.

Math Problem Statement

( k − 3 ) x + 9

Solution

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