The question asks to find the discriminant and the number of real roots for the quadratic equation:

$4x^2 + 12x + 9 = 0$

To solve this, we use the discriminant formula, which is part of the quadratic formula:

$\Delta = b^2 - 4ac$

For the equation $4x^2 + 12x + 9 = 0$, the coefficients are:

• $a = 4$
• $b = 12$
• $c = 9$

Now, calculate the discriminant:

$\Delta = (12)^2 - 4(4)(9)$ $\Delta = 144 - 144$ $\Delta = 0$

Since the discriminant is 0, there is exactly one real root.

Therefore, the correct answer is A: 0; one real root.

Do you want more details on how this works or have any other questions?

Here are 5 related questions to expand on this concept:

1. What happens if the discriminant is negative?
2. How do you factor a quadratic equation with a discriminant of zero?
3. Can a quadratic equation have two complex roots if the discriminant is positive?
4. What is the geometric significance of having only one real root for a quadratic equation?
5. How does changing the value of $c$ affect the number of real roots?

Tip: The discriminant is the key to quickly determining the nature of the roots of any quadratic equation.