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)   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
\[\text{B}\]

To determine which of the given equations are true identities, we'll examine both equations individually.

### Equation A:
\[
(5b-2)^2 + 4 = 25b^2 - 20b
\]

First, let's expand \((5b-2)^2\):
\[
(5b-2)^2 = 25b^2 - 20b + 4
\]

Now, substitute this back into the equation:
\[
25b^2 - 20b + 4 + 4 = 25b^2 - 20b
\]
\[
25b^2 - 20b + 8 = 25b^2 - 20b
\]

Clearly, \( 25b^2 - 20b + 8 \neq 25b^2 - 20b \), so **Equation A is not a true identity**.

### Equation B:
\[
(2x^2 + y^2)(2x^2 - y^2) = 4x^4 - y^4
\]

This is a difference of squares on the left-hand side:
\[
(2x^2 + y^2)(2x^2 - y^2) = (2x^2)^2 - (y^2)^2
\]

Simplify:
\[
(2x^2)^2 - (y^2)^2 = 4x^4 - y^4
\]

So, **Equation B is a true identity**.

### Conclusion:
Only Equation B is a true identity. The correct choice is **B**.

---

Would you like more details on how I expanded and simplified the equations? Here are 5 related questions for further exploration:

1. How do you expand binomials such as \((a - b)^2\)?
2. What are the key properties of a difference of squares?
3. How do you verify if an equation is an identity?
4. Can a cubic equation ever be a true identity?
5. What role do identities play in simplifying complex algebraic expressions?

**Tip**: Always look for common algebraic patterns, like difference of squares or perfect square trinomials, to simplify expressions effectively.

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:
(Choice A) Only \[\text{A}\]
(Choice B) Only \[\text{B}\]
(Choice C) Both \[\text{A}\] and \[\text{B}\]
(Choice D) Neither \[\text{A}\] nor \[\text{B}\]

Explore how to determine true identities in polynomial equations, focusing on expanding binomials and using the difference of squares theorem. This problem is suitable for students in Grades 9-10, providing a step-by-step breakdown of each equation.

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