Question
Which expressions are equivalent to 12x
2 − 48x + 48?
Select TWO correct answers.
(2 points)
12(x − 2)
2
−12 x
2 ( + 4x + 4)
12(x − 4)(x − 1)
−12(x + 2)
2
12 x
2 ( − 4x + 4)
2 − 48x + 48?
Select TWO correct answers.
(2 points)
12(x − 2)
2
−12 x
2 ( + 4x + 4)
12(x − 4)(x − 1)
−12(x + 2)
2
12 x
2 ( − 4x + 4)
Answers
GPT-4o mini
To determine which expressions are equivalent to \( 12x^2 - 48x + 48 \), we can first factor the original expression.
1. First, factor out the greatest common factor from \( 12x^2 - 48x + 48 \):
\[
12(x^2 - 4x + 4)
\]
2. Next, we can factor the quadratic expression \( x^2 - 4x + 4 \):
\[
x^2 - 4x + 4 = (x - 2)^2
\]
3. Therefore, we can rewrite the entire expression as:
\[
12(x - 2)^2
\]
Now we check the provided expressions to see which ones are equivalent:
1. **\( 12(x - 2)^2 \)** - This is equivalent to the simplified factorization we found.
2. **\( -12(x + 2)^2 \)** - This is not equivalent because it has a negative sign and a different expression in the parentheses.
3. **\( 12(x - 4)(x - 1) \)** - This does not match the original expression as the roots differ.
4. **\( 12x^2(-4x + 4) \)** - This does not match either; the overall degree and form change.
5. **\( -12x^2 + 48x - 48 \)** - This is also not equivalent because it negates and changes the signs.
It appears that the only equivalent choice is the expressible factored form itself.
So **the two correct answers** (that simply match equivalent expressions) are:
1. \( 12(x − 2)^2 \)
2. The expressions \( -12(x + 2)^2 \) and \( 12 x^2(-4x + 4) \) also do not meet standard factoring so no other options suffice.
If you list `12(x − 2)^2` and similar matches as default options based on the listed closest equivalents. None of the given alternatives correctly articulate the original polynomial form based on standard algebraic principles.
However, I can verify exact answers if you attempt to recheck them.
1. First, factor out the greatest common factor from \( 12x^2 - 48x + 48 \):
\[
12(x^2 - 4x + 4)
\]
2. Next, we can factor the quadratic expression \( x^2 - 4x + 4 \):
\[
x^2 - 4x + 4 = (x - 2)^2
\]
3. Therefore, we can rewrite the entire expression as:
\[
12(x - 2)^2
\]
Now we check the provided expressions to see which ones are equivalent:
1. **\( 12(x - 2)^2 \)** - This is equivalent to the simplified factorization we found.
2. **\( -12(x + 2)^2 \)** - This is not equivalent because it has a negative sign and a different expression in the parentheses.
3. **\( 12(x - 4)(x - 1) \)** - This does not match the original expression as the roots differ.
4. **\( 12x^2(-4x + 4) \)** - This does not match either; the overall degree and form change.
5. **\( -12x^2 + 48x - 48 \)** - This is also not equivalent because it negates and changes the signs.
It appears that the only equivalent choice is the expressible factored form itself.
So **the two correct answers** (that simply match equivalent expressions) are:
1. \( 12(x − 2)^2 \)
2. The expressions \( -12(x + 2)^2 \) and \( 12 x^2(-4x + 4) \) also do not meet standard factoring so no other options suffice.
If you list `12(x − 2)^2` and similar matches as default options based on the listed closest equivalents. None of the given alternatives correctly articulate the original polynomial form based on standard algebraic principles.
However, I can verify exact answers if you attempt to recheck them.