determine the nature of the roots of 3x^2+4x+k=0 by solving for k

The nature of the roots of the quadratic equation \( 3x^2 + 4x + k = 0 \) depends on the discriminant (\( \Delta \)) of the equation, which is given by the formula:

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

For the equation \( 3x^2 + 4x + k = 0 \), we identify:

- \( a = 3 \)
- \( b = 4 \)
- \( c = k \)

Substituting into the discriminant formula:

\[
\Delta = 4^2 - 4(3)(k) = 16 - 12k
\]

Now, based on the value of \( \Delta \), we can determine the nature of the roots:

- If \( \Delta > 0 \), the equation has **two distinct real roots**.
- If \( \Delta = 0 \), the equation has **one real root** (repeated root).
- If \( \Delta < 0 \), the equation has **two complex roots**.

So, to determine the nature of the roots in terms of \( k \):

1. **Two distinct real roots**: when \( \Delta > 0 \), i.e.
   \[
   16 - 12k > 0 \quad \Rightarrow \quad k < \frac{4}{3}
   \]

2. **One real root**: when \( \Delta = 0 \), i.e.
   \[
   16 - 12k = 0 \quad \Rightarrow \quad k = \frac{4}{3}
   \]

3. **Two complex roots**: when \( \Delta < 0 \), i.e.
   \[
   16 - 12k < 0 \quad \Rightarrow \quad k > \frac{4}{3}
   \]

### Summary:
- If \( k < \frac{4}{3} \), the roots are **two distinct real numbers**.
- If \( k = \frac{4}{3} \), there is **one real repeated root**.
- If \( k > \frac{4}{3} \), the roots are **two complex numbers**.

Would you like more details or have any questions?

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Here are 5 related questions to explore further:
1. What happens if \( k \) is exactly zero in the quadratic equation?
2. How does the discriminant change if the quadratic equation has different coefficients?
3. How can you find the exact roots of the equation for specific values of \( k \)?
4. What is the geometric interpretation of complex roots on a graph?
5. Can you provide an example of a quadratic equation where the discriminant is negative?

**Tip**: Always check the discriminant first to quickly assess the type of solutions without solving the quadratic equation!

Determine the nature of the roots of 3x^2 + 4x + k = 0 by solving for k

Learn how to determine the nature of the roots of the quadratic equation 3x^2 + 4x + k = 0 by using the discriminant. Explore algebraic concepts such as the discriminant formula and how it defines whether roots are real or complex. This guide is suitable for students in Grades 8-10.

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