3x^2+4x+k=0 determine the value of k and the nature of the roots

To determine the value of \( k \) and the nature of the roots for the quadratic equation:

\[
3x^2 + 4x + k = 0
\]

### Step 1: Use the Discriminant Formula
The discriminant (\( \Delta \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

Here, \( a = 3 \), \( b = 4 \), and \( c = k \).

Thus, the discriminant becomes:

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

### Step 2: Analyze the Nature of the Roots
- **If \( \Delta > 0 \)**: The quadratic equation has two distinct real roots.
- **If \( \Delta = 0 \)**: The quadratic equation has exactly one real root (repeated root).
- **If \( \Delta < 0 \)**: The quadratic equation has two complex (non-real) roots.

### Step 3: Determine \( k \)
You didn't provide a specific condition for the nature of the roots, so we can solve for \( k \) based on different scenarios:

1. **For real and equal roots** (\( \Delta = 0 \)):  
   Set \( \Delta = 0 \):
   \[
   16 - 12k = 0 \implies k = \frac{16}{12} = \frac{4}{3}
   \]

2. **For real and distinct roots** (\( \Delta > 0 \)):  
   Set \( \Delta > 0 \):
   \[
   16 - 12k > 0 \implies 16 > 12k \implies k < \frac{4}{3}
   \]

3. **For complex roots** (\( \Delta < 0 \)):  
   Set \( \Delta < 0 \):
   \[
   16 - 12k < 0 \implies 16 < 12k \implies k > \frac{4}{3}
   \]

### Conclusion
- \( k = \frac{4}{3} \) leads to one real repeated root.
- \( k < \frac{4}{3} \) leads to two distinct real roots.
- \( k > \frac{4}{3} \) leads to two complex roots.

Would you like more details on a specific case or have any questions?

Here are 5 related questions to explore:
1. What happens to the quadratic equation if \( k = 0 \)?
2. How do you find the actual roots of the equation for specific values of \( k \)?
3. What is the geometric interpretation of real and complex roots?
4. Can you apply the quadratic formula to find the roots for \( k = 1 \)?
5. How do different values of \( k \) affect the graph of \( 3x^2 + 4x + k = 0 \)?

**Tip**: Always check the discriminant first to determine the type of roots before solving the quadratic equation.

3x^2 + 4x + k = 0 determine the value of k and the nature of the roots

Explore how to determine the value of k and analyze the nature of roots for the quadratic equation 3x^2 + 4x + k = 0. This guide covers essential algebraic concepts, including the discriminant formula, and provides a step-by-step solution. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation