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The question asks to find the value of the constant \( k \) for which the equation:

\[
2x^2 + (k + 3)x + 2k = 0
\]

has equal roots.

To determine when a quadratic equation has equal roots, we need to use the discriminant, \( \Delta \), of a quadratic equation in the form \( ax^2 + bx + c = 0 \). For equal roots, the discriminant must be zero:

\[
\Delta = b^2 - 4ac = 0
\]

### Step 1: Identify \( a \), \( b \), and \( c \)
In this equation:
- \( a = 2 \)
- \( b = k + 3 \)
- \( c = 2k \)

### Step 2: Set up the discriminant equation
Using \( \Delta = b^2 - 4ac = 0 \):

\[
(k + 3)^2 - 4 \cdot 2 \cdot 2k = 0
\]

### Step 3: Expand and simplify
1. Expand \( (k + 3)^2 \):
   \[
   (k + 3)^2 = k^2 + 6k + 9
   \]

2. Calculate \( 4 \cdot 2 \cdot 2k \):
   \[
   4 \cdot 2 \cdot 2k = 16k
   \]

3. Substitute back into the discriminant equation:
   \[
   k^2 + 6k + 9 - 16k = 0
   \]

4. Simplify the equation:
   \[
   k^2 - 10k + 9 = 0
   \]

### Step 4: Solve the quadratic equation
Now we solve \( k^2 - 10k + 9 = 0 \) using the quadratic formula:

\[
k = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}
\]

\[
k = \frac{10 \pm \sqrt{100 - 36}}{2}
\]

\[
k = \frac{10 \pm \sqrt{64}}{2}
\]

\[
k = \frac{10 \pm 8}{2}
\]

This gives two solutions for \( k \):
1. \( k = \frac{10 + 8}{2} = 9 \)
2. \( k = \frac{10 - 8}{2} = 1 \)

### Final Answer
The values of \( k \) for which the equation has equal roots are:

\[
k = 9 \quad \text{or} \quad k = 1
\]

Would you like further details or have any questions?

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Here are 5 related questions that expand on this topic:

1. What does it mean for a quadratic equation to have equal roots?
2. How can you find the roots of a quadratic equation if the discriminant is not zero?
3. Why is the discriminant \( b^2 - 4ac \) crucial for determining the nature of the roots?
4. Can there be any values of \( k \) that result in imaginary roots for this equation?
5. How would the solution change if the coefficient of \( x^2 \) was not 2 but another number?

**Tip:** Remember, if the discriminant of a quadratic equation is zero, it guarantees that both roots are equal (i.e., a perfect square solution).

Find the value of the constant k for which the equation 2x^2 + (k + 3)x + 2k = 0 has equal roots.

Discover how to find the constant k for which the quadratic equation 2x^2 + (k + 3)x + 2k = 0 has equal roots. This solution explains the use of the discriminant formula and the conditions for equal roots, ideal for high school algebra students.

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