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?

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).