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Let's analyze the problem step by step:

### Step 1: Analyze the first equation
The first equation is:
\[
2x^2 + kx + 5 = 0
\]
It is given that this equation **has no real roots**. For a quadratic equation \(ax^2 + bx + c = 0\), the condition for having no real roots is that its discriminant must be negative. The discriminant \(\Delta\) is given by:
\[
\Delta = b^2 - 4ac
\]
For the given equation:
- \(a = 2\)
- \(b = k\)
- \(c = 5\)

So the discriminant is:
\[
\Delta = k^2 - 4(2)(5) = k^2 - 40
\]
For the equation to have no real roots, we require:
\[
k^2 - 40 < 0
\]
This simplifies to:
\[
k^2 < 40
\]
Taking the square root on both sides:
\[
-\sqrt{40} < k < \sqrt{40}
\]
Since \(k\) is an integer, we consider the integer values within this range. The square root of 40 is approximately 6.32, so:
\[
-6 < k < 6
\]
Thus, \(k\) can take integer values from \(-6\) to \(5\) inclusive.

### Step 2: Analyze the second equation
The second equation is:
\[
x^2 + (k-5)x + 1 = 0
\]
It is given that this equation **has two distinct real roots**. For a quadratic equation to have two distinct real roots, the discriminant must be positive. The discriminant for this equation is:
\[
\Delta = (k-5)^2 - 4(1)(1) = (k-5)^2 - 4
\]
For the equation to have two distinct real roots, we require:
\[
(k-5)^2 - 4 > 0
\]
This simplifies to:
\[
(k-5)^2 > 4
\]
Taking the square root:
\[
|k-5| > 2
\]
This leads to two possible cases:
1. \(k - 5 > 2 \) which simplifies to \(k > 7\)
2. \(k - 5 < -2\) which simplifies to \(k < 3\)

So, \(k\) must either be \(k > 7\) or \(k < 3\).

### Step 3: Combine the conditions from both equations
From Step 1, \(k\) is between \(-6\) and \(5\) inclusive. From Step 2, \(k\) must be either \(k < 3\) or \(k > 7\).

Combining these conditions:
- \(k\) values satisfying both conditions are those where \(k < 3\).
- \(k\) can therefore take the values \(-6, -5, -4, -3, -2, -1, 0, 1, 2\).

Thus, there are **9 possible values** for \(k\) that satisfy both conditions.

### Answer:
The number of possible values of \(k\) is **9**.

Would you like more details on any specific part of this solution?

Here are 5 related questions you might be interested in:
1. How do we derive the formula for the discriminant of a quadratic equation?
2. What are the implications of having a discriminant equal to zero?
3. Can we have an equation with complex roots if the discriminant is positive?
4. How would the solution change if the second equation was required to have a double root?
5. How does the range of values for \(k\) change if we alter the coefficients of the quadratic equations?

**Tip:** Understanding how the discriminant affects the nature of the roots is crucial in solving quadratic equations.

Suppose k is any integer such that the equation 2x^2 + kx + 5 = 0 has no real roots and the equation x^2 + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is?

Learn how to solve a problem involving two quadratic equations, discriminants, and integer values of k. This problem covers key concepts in algebra and quadratic equations, focusing on conditions for real and distinct roots. Suitable for students in Grades 9-12.

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