IF both the roots of the quadratic equation `x^2-2kx+k^2 +k -5=0` are less than 5, then k lies in the interval

Options

A. (5, 6]
B. `(6, oo)`
C. `(-oo, 4)`
D. [4,5]

Answer

To solve the problem, we need to determine the values of \( k \) such that both roots of the quadratic equation \( x^2 - 2kx + (k^2 + k - 5) = 0 \) are less than 5.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = -2k \), and \( c = k^2 + k - 5 \).

2. **Use the quadratic formula**:
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{2k \pm \sqrt{(-2k)^2 - 4 \cdot 1 \cdot (k^2 + k - 5)}}{2 \cdot 1} \]
Simplifying this gives:
\[ x = \frac{2k \pm \sqrt{4k^2 - 4(k^2 + k - 5)}}{2} \]

3. **Simplify the expression under the square root**:
\[ x = \frac{2k \pm \sqrt{4k^2 - 4k^2 - 4k + 20}}{2} \]
This simplifies to:
\[ x = \frac{2k \pm \sqrt{-4k + 20}}{2} \]
Further simplifying gives:
\[ x = k \pm \sqrt{5 - k} \]

4. **Set the condition for the roots**:
We need both roots to be less than 5:
\[ k - \sqrt{5 - k} < 5 \quad \text{and} \quad k + \sqrt{5 - k} < 5 \]

5. **Solve the first inequality**:
\[ k - \sqrt{5 - k} < 5 \]
Rearranging gives:
\[ -\sqrt{5 - k} < 5 - k \]
Squaring both sides (noting that both sides are positive):
\[ 5 - k > (5 - k)^2 \]
Expanding the right side:
\[ 5 - k > 25 - 10k + k^2 \]
Rearranging gives:
\[ k^2 - 9k + 20 > 0 \]

6. **Factor the quadratic**:
Factoring gives:
\[ (k - 4)(k - 5) > 0 \]
The critical points are \( k = 4 \) and \( k = 5 \).

7. **Determine the intervals**:
Testing intervals:
- For \( k < 4 \): Both factors are negative, product is positive.
- For \( 4 < k < 5 \): One factor is negative, product is negative.
- For \( k > 5 \): Both factors are positive, product is positive.

Thus, the solution for this inequality is:
\[ k < 4 \quad \text{or} \quad k > 5 \]

8. **Solve the second inequality**:
\[ k + \sqrt{5 - k} < 5 \]
Rearranging gives:
\[ \sqrt{5 - k} < 5 - k \]
Squaring both sides:
\[ 5 - k < (5 - k)^2 \]
This leads to the same quadratic inequality as before:
\[ k^2 - 9k + 20 > 0 \]
Thus, the solution remains the same.

9. **Final interval for \( k \)**:
The values of \( k \) that satisfy both conditions are:
\[ k < 4 \]
Therefore, the interval for \( k \) is:
\[ k \in (-\infty, 4) \]

### Conclusion:
The correct interval for \( k \) is \( (-\infty, 4) \).