Determine the discriminant and nature of 2x²+6x+4=0

To determine the discriminant and the nature of the roots for the quadratic equation \(2x^2 + 6x + 4 = 0\), we can follow these steps:

### 1. Identify the coefficients:
The quadratic equation is of the form \(ax^2 + bx + c = 0\), where:
- \(a = 2\)
- \(b = 6\)
- \(c = 4\)

### 2. Calculate the discriminant:
The discriminant (\(\Delta\)) is given by the formula:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\):
\[
\Delta = 6^2 - 4(2)(4)
\]
\[
\Delta = 36 - 32
\]
\[
\Delta = 4
\]

### 3. Determine the nature of the roots:
The discriminant helps us understand the nature of the roots:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has exactly one real root (a repeated root).
- If \(\Delta < 0\), the equation has two complex roots.

Since \(\Delta = 4\), which is greater than 0, the equation \(2x^2 + 6x + 4 = 0\) has **two distinct real roots**.

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Do you have any questions or would you like more details on any part of the solution? Here are some related questions you might find interesting:

1. How do you find the roots of a quadratic equation using the quadratic formula?
2. What is the geometric interpretation of the discriminant?
3. Can you derive the quadratic formula from the standard form of a quadratic equation?
4. How do you solve a quadratic equation by completing the square?
5. What happens to the roots of a quadratic equation if the discriminant is negative?

**Tip:** Always check the discriminant first when solving a quadratic equation to quickly determine the nature of the roots.

Explore how to determine the discriminant and the nature of roots for the quadratic equation 2x² + 6x + 4 = 0. This solution walks through key algebraic concepts, including calculating the discriminant (Δ = 4) and interpreting the two distinct real roots of the equation. Suitable for students in Grades 9-10.

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