To determine the number of distinct roots for the equation [tex]\( y = (3x - 6)^2 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make [tex]\( y \)[/tex] equal to zero.
1. Setting the equation to zero:
We start by setting [tex]\( y \)[/tex] to zero:
[tex]\[
(3x - 6)^2 = 0
\][/tex]
2. Solving the equation:
To solve this, we can take the square root of both sides. The square root of zero is still zero:
[tex]\[
3x - 6 = 0
\][/tex]
3. Isolating [tex]\( x \)[/tex]:
Now, we solve for [tex]\( x \)[/tex] by isolating it on one side of the equation. First, add 6 to both sides:
[tex]\[
3x = 6
\][/tex]
Next, divide both sides by 3:
[tex]\[
x = \frac{6}{3}
\][/tex]
Simplifying that gives:
[tex]\[
x = 2
\][/tex]
Since squaring a number yields a single root when the result is zero, [tex]\( x = 2 \)[/tex] is the only root for the equation [tex]\( (3x - 6)^2 = 0 \)[/tex]. Hence, the equation has exactly one distinct root.
Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
