The given quadratic equation is:

$kx^2 + 2x + 3 = 0$

For the roots of a quadratic equation to be equal, the discriminant (Δ) must be zero. The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

For the equation $kx^2 + 2x + 3 = 0$, the coefficients are:

• $a = k$
• $b = 2$
• $c = 3$

Substituting these values into the discriminant formula:

$\Delta = 2^2 - 4(k)(3)$

$\Delta = 4 - 12k$

For equal roots, $\Delta = 0$, so:

$4 - 12k = 0$

Solving for $k$:

$12k = 4$

$k = \frac{4}{12} = \frac{1}{3}$

Therefore, the value of $k$ for which the roots of the equation are equal is:

$k = \frac{1}{3}$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the condition for real and distinct roots in a quadratic equation?
2. How do you solve a quadratic equation using the quadratic formula?
3. Can you explain how the discriminant affects the nature of the roots?
4. What happens to the graph of a quadratic function when the roots are equal?
5. How would you solve the quadratic equation for different values of $k$?

Tip: For a quadratic equation, if the discriminant is positive, the roots are real and distinct; if it's zero, the roots are real and equal; if it's negative, the roots are complex.