The discriminant is a key concept in the study of quadratic equations, providing valuable information about the nature of their roots. For a standard quadratic equation of the form ax² + bx + c = 0, the discriminant (often denoted as D) is calculated using the formula:
- D = b² – 4ac
In the case of the given equation, 3x² + 6x + 2, we can identify the coefficients:
- a = 3
- b = 6
- c = 2
Now, substituting these values into the discriminant formula:
- D = 6² – 4(3)(2)
Calculating 6² gives us 36, and 4(3)(2) equals 24. So, we can continue with our calculation:
- D = 36 – 24
This simplifies to:
- D = 12
Now, what does this discriminant value tell us? Since the discriminant D = 12 is greater than zero, we can conclude that:
- The quadratic equation has two distinct real roots.
This information can be incredibly useful when analyzing the graph of the quadratic function or when solving for the roots using methods such as the quadratic formula.
In summary, the discriminant of the quadratic equation 3x² + 6x + 2 is 12, indicating that this equation will have two distinct real roots.