To determine the value(s) of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex], follow these steps:
1. Identify the coefficients: The given quadratic equation is [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex]. Here, the coefficients are [tex]\( a = 4 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = 16 \)[/tex].
2. Calculate the discriminant ([tex]\( D \)[/tex]) of the quadratic equation:
[tex]\[
D = b^2 - 4ac
\][/tex]
Plugging in the values:
[tex]\[
D = (-16)^2 - 4 \cdot 4 \cdot 16
\][/tex]
[tex]\[
D = 256 - 256
\][/tex]
[tex]\[
D = 0
\][/tex]
Since the discriminant is zero, there is exactly one unique real solution (a repeated root).
3. Use the quadratic formula to find the root(s). The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
Given that [tex]\( D = 0 \)[/tex], the formula simplifies to:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
Substitute the coefficients [tex]\( a = 4 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = 16 \)[/tex]:
[tex]\[
x = \frac{-(-16)}{2 \cdot 4}
\][/tex]
[tex]\[
x = \frac{16}{8}
\][/tex]
[tex]\[
x = 2
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 4x^2 - 16x + 16 = 0 \)[/tex] is [tex]\( x = 2 \)[/tex].
To summarize, the correct value is [tex]\(\boxed{2}\)[/tex].
