the value of k for which the roots of equation kx ^ 2 + 2x + 3 = 0 are equal is

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.

The value of k for which the roots of equation kx^2 + 2x + 3 = 0 are equal is

Discover how to find the value of k for which the roots of the quadratic equation kx^2 + 2x + 3 = 0 are equal. This step-by-step solution covers key algebraic concepts like the discriminant and its role in determining the nature of roots. Suitable for students in Grades 9-12.

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