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.