A quadratic equation is given by:
2x^2+3x+4=0.
Determine the discriminant d and explain what this number tells us about the number of solutions to the equation.

To determine the discriminant \( d \) of a quadratic equation, we use the formula:

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

For the quadratic equation \( 2x^2 + 3x + 4 = 0 \), the coefficients are:

- \( a = 2 \)
- \( b = 3 \)
- \( c = 4 \)

Now, substitute these values into the discriminant formula:

\[
d = (3)^2 - 4(2)(4)
\]
\[
d = 9 - 32
\]
\[
d = -23
\]

### Interpretation of the Discriminant

The discriminant \( d = -23 \) is negative. The value of the discriminant tells us the nature of the roots (solutions) of the quadratic equation:

- **If \( d > 0 \)**: The equation has two distinct real roots.
- **If \( d = 0 \)**: The equation has exactly one real root (a repeated root).
- **If \( d < 0 \)**: The equation has two complex (non-real) roots.

Since \( d = -23 \), which is less than 0, the quadratic equation \( 2x^2 + 3x + 4 = 0 \) has **two complex (non-real) roots**.

Would you like more details or have any questions about this? Here are some related questions:

1. How do you calculate the roots of a quadratic equation when the discriminant is negative?
2. Can you provide examples of quadratic equations with a positive discriminant?
3. What does it mean geometrically when the discriminant is negative?
4. How can the discriminant help in solving real-world problems?
5. What happens if the discriminant is exactly zero?

**Tip:** When solving quadratic equations, always check the discriminant first to determine the nature of the roots before attempting to solve the equation.

Explore how to determine the discriminant of the quadratic equation 2x^2 + 3x + 4 = 0 and what it reveals about the number of solutions. Learn about quadratic equations, discriminant calculation, and the interpretation of different discriminant values. Suitable for Grades 10-12 students.

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