**Which of the equations are true identities?**

\[\begin{aligned}
\text{A. }&(5b-2)^2+4=25b^2-20b
\\\\
\text{B. }&(2x^2+y^2)(2x^2-y^2)=4x^2-y^2
\end{aligned}\]

Choose 1 answer:

Choose 1 answer:

(Choice A)   

Only \[\text{A}\]

A

Only \[\text{A}\]

(Choice B)   

Only \[\text{B}\]

B

Only \[\text{B}\]

(Choice C, Checked)   

Both \[\text{A}\] and \[\text{B}\]

C

Both \[\text{A}\] and \[\text{B}\]

(Choice D)   

Neither \[\text{A}\] nor \[\text{B}\]

D

Neither \[\text{A}\] nor \[\text{B}\]

Let's evaluate each equation separately to determine which are true identities.

### Equation A:
\[
(5b - 2)^2 + 4 = 25b^2 - 20b
\]
1. Expand the left-hand side:
\[
(5b - 2)^2 = (5b)^2 - 2 \cdot (5b) \cdot 2 + (-2)^2 = 25b^2 - 20b + 4
\]
So:
\[
25b^2 - 20b + 4 + 4 = 25b^2 - 20b + 4
\]
This simplifies to:
\[
25b^2 - 20b + 4 \neq 25b^2 - 20b
\]
Therefore, **Equation A is not an identity.**

### Equation B:
\[
(2x^2 + y^2)(2x^2 - y^2) = 4x^4 - y^4
\]
The left-hand side is a difference of squares:
\[
(2x^2 + y^2)(2x^2 - y^2) = (2x^2)^2 - (y^2)^2 = 4x^4 - y^4
\]
The left-hand side simplifies exactly to the right-hand side, so **Equation B is a true identity.**

### Conclusion:
The correct answer is **(Choice B: Only Equation B is a true identity).**

Would you like more details or have any questions?

Here are some related questions:
1. What is a true mathematical identity?
2. How do you expand binomials using algebraic identities?
3. What are some common cases of the difference of squares?
4. How can you verify if an equation is an identity?
5. Can a false identity still have solutions for specific values?

**Tip:** Always check if both sides of an equation simplify to the same expression to verify if it's an identity.

Which of the equations are true identities?

\[
\begin{aligned}
\text{A. }&(5b-2)^2+4=25b^2-20b
\\
\text{B. }&(2x^2+y^2)(2x^2-y^2)=4x^4-y^4
\end{aligned}
\]

Choose 1 answer:

(Choice A) Only \(A\)
(Choice B) Only \(B\)
(Choice C) Both \(A\) and \(B\)
(Choice D) Neither \(A\) nor \(B\)

Explore whether given algebraic equations are true identities by applying binomial expansion and the difference of squares method. This problem covers foundational concepts in algebra, suitable for students in Grades 9-12, and guides learners through verifying identities step-by-step.

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