4x^2+12x+9=0 can you solve this problem with this formula D=b^2-4ac

To solve the quadratic equation \( 4x^2 + 12x + 9 = 0 \) using the discriminant formula \( D = b^2 - 4ac \), let's go through the steps:

### 1. Identify the coefficients:
For the equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 4 \) (coefficient of \( x^2 \))
- \( b = 12 \) (coefficient of \( x \))
- \( c = 9 \) (constant term)

### 2. Calculate the discriminant \( D \):
\[
D = b^2 - 4ac
\]
Substitute the values of \( a \), \( b \), and \( c \):
\[
D = 12^2 - 4(4)(9) = 144 - 144 = 0
\]

### 3. Interpret the discriminant:
Since \( D = 0 \), the equation has exactly one real root, also known as a repeated or double root.

### 4. Solve for the root using the quadratic formula:
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
Substitute the values of \( b \), \( D \), and \( a \):
\[
x = \frac{-12 \pm \sqrt{0}}{2(4)} = \frac{-12 \pm 0}{8} = \frac{-12}{8} = -\frac{3}{2}
\]

### 5. Final Answer:
The equation has one real root:
\[
x = -\frac{3}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. What would the discriminant \( D \) be if the constant term \( c \) were different?
2. How would you factorize the quadratic equation \( 4x^2 + 12x + 9 = 0 \)?
3. Can you graph the equation and identify the point where it touches the x-axis?
4. What does it mean geometrically when a quadratic equation has a double root?
5. How can you use the quadratic formula to solve equations where the discriminant is negative?

**Tip:** When the discriminant \( D = 0 \), it means the quadratic equation has a perfect square trinomial, leading to a single root.

Learn how to solve the quadratic equation 4x^2 + 12x + 9 = 0 using the discriminant formula D = b^2 - 4ac. This problem covers key algebraic concepts such as the quadratic equation formula and the discriminant. Suitable for students in Grades 10-12.

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