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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.

Find the discriminant and the number of real roots for the equation 4x^2 + 12x + 9 = 0.

Solve the quadratic equation 4x^2 + 12x + 9 = 0 by finding its discriminant (Δ) and determining the number of real roots. Learn step-by-step to use the formula Δ = b^2 - 4ac, where Δ = 0 means one real root. Suitable for Grades 8-10.

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