To determine which expressions are equivalent to the quadratic expression \( 12x^2 - 48x + 48 \), we can factor the original expression or manipulate the given options.
First, we can factor \( 12x^2 - 48x + 48 \):
-
Factor out 12 from the expression: \[ 12(x^2 - 4x + 4) \]
-
Now, \( x^2 - 4x + 4 \) can be factored further as: \[ x^2 - 4x + 4 = (x - 2)^2 \]
So we have: \[ 12(x^2 - 4x + 4) = 12(x - 2)^2 \]
From this, we find that:
- The expression \( 12(x - 2)^2 \) is indeed equivalent to \( 12x^2 - 48x + 48 \).
- The expression \( 12(x^2 - 4x + 4) \) is just a different representation and is also equivalent to the original expression.
Now we can analyze the answer choices:
- 12(x – 2)²: Equivalent (as we derived).
- 12(x + 2)²: Not equivalent.
- 12(x² ‒ 4x + 4): Equivalent (as we derived).
- 12(x² ‒ 6x + 6): Not equivalent.
Therefore, the two correct answers are:
- 12(x – 2)²
- 12(x² ‒ 4x + 4)