To find the discriminant and the number of real roots for the quadratic equation [tex]\(4x^2 + 16x + 16 = 0\)[/tex], we follow these steps:
### Step 1: Identify the coefficients
The quadratic equation in standard form is [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, we can identify:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 16\)[/tex]
- [tex]\(c = 16\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2+bx+c=0\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[
\Delta = 16^2 - 4 \cdot 4 \cdot 16
\][/tex]
Calculate step-by-step:
[tex]\[
16^2 = 256
\][/tex]
[tex]\[
4 \cdot 4 = 16
\][/tex]
[tex]\[
16 \cdot 16 = 256
\][/tex]
[tex]\[
\Delta = 256 - 256 = 0
\][/tex]
So, the discriminant [tex]\(\Delta\)[/tex] is [tex]\(0\)[/tex].
### Step 3: Determine the number of real roots
The number of real roots of a quadratic equation depends on the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real root (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], there are no real roots (the roots are complex).
In this case, since [tex]\(\Delta = 0\)[/tex], there is exactly one real root.
### Step 4: Choose the correct option
From the given options:
- A. -256 ; one real root
- B. -256 ; no real roots
- C. 0; two real roots
- D. 0; one real root
The correct option based on our calculations is:
[tex]\[
\boxed{\text{D. } 0; \text{ one real root}}
\][/tex]
