Question
determine if the expression shown can be factored using differences of two squares 50x^4-25y^2
Will It be
Difference of two squares cannot be used because there are more than two terms
The expression can be factored using differences of two squares
Differences of two squares cannot be used because all terms are not perfect
Differences of two squares cannot be used because two terms are being added together
Will It be
Difference of two squares cannot be used because there are more than two terms
The expression can be factored using differences of two squares
Differences of two squares cannot be used because all terms are not perfect
Differences of two squares cannot be used because two terms are being added together
Answers
GPT-4o mini
To determine whether the expression \( 50x^4 - 25y^2 \) can be factored using the difference of two squares, we first rewrite the expression in a form that highlights a potential difference of squares.
1. **Factor out the greatest common factor (GCF)**:
\[
50x^4 - 25y^2 = 25(2x^4 - y^2)
\]
2. **Now, look at the expression inside the parentheses**:
\[
2x^4 - y^2
\]
The difference of squares formula states that \( a^2 - b^2 = (a - b)(a + b) \).
To apply this, we notice that \( 2x^4 \) is not itself a perfect square since \( 2 \) is not a perfect square. However, we can rewrite \( 2x^4 \) as \( \sqrt{2}x^2 \) so that:
\[
a = \sqrt{2}x^2 \quad \text{and} \quad b = y
\]
Given that, we can see that we have the form \( a^2 - b^2 \) which implies:
\[
( \sqrt{2}x^2 - y)( \sqrt{2}x^2 + y)
\]
Thus, we can factor the expression as:
\[
25( \sqrt{2}x^2 - y)( \sqrt{2}x^2 + y)
\]
Now, let's evaluate the provided statements:
1. "Difference of two squares cannot be used because there are more than two terms" - **False**. There are two terms after factoring out the GCF.
2. "The expression can be factored using differences of two squares" - **True**. We can factor \( 2x^4 - y^2 \) as described above.
3. "Differences of two squares cannot be used because all terms are not perfect" - **False**. We can still apply the difference of squares to the expression \( 2x^4 - y^2 \) as shown.
4. "Differences of two squares cannot be used because two terms are being added together" - **False**. This statement refers to the form of a difference but is not relevant after factoring the GCF.
Therefore, the correct conclusion is:
**The expression can be factored using differences of two squares.**
1. **Factor out the greatest common factor (GCF)**:
\[
50x^4 - 25y^2 = 25(2x^4 - y^2)
\]
2. **Now, look at the expression inside the parentheses**:
\[
2x^4 - y^2
\]
The difference of squares formula states that \( a^2 - b^2 = (a - b)(a + b) \).
To apply this, we notice that \( 2x^4 \) is not itself a perfect square since \( 2 \) is not a perfect square. However, we can rewrite \( 2x^4 \) as \( \sqrt{2}x^2 \) so that:
\[
a = \sqrt{2}x^2 \quad \text{and} \quad b = y
\]
Given that, we can see that we have the form \( a^2 - b^2 \) which implies:
\[
( \sqrt{2}x^2 - y)( \sqrt{2}x^2 + y)
\]
Thus, we can factor the expression as:
\[
25( \sqrt{2}x^2 - y)( \sqrt{2}x^2 + y)
\]
Now, let's evaluate the provided statements:
1. "Difference of two squares cannot be used because there are more than two terms" - **False**. There are two terms after factoring out the GCF.
2. "The expression can be factored using differences of two squares" - **True**. We can factor \( 2x^4 - y^2 \) as described above.
3. "Differences of two squares cannot be used because all terms are not perfect" - **False**. We can still apply the difference of squares to the expression \( 2x^4 - y^2 \) as shown.
4. "Differences of two squares cannot be used because two terms are being added together" - **False**. This statement refers to the form of a difference but is not relevant after factoring the GCF.
Therefore, the correct conclusion is:
**The expression can be factored using differences of two squares.**