Math Problem Statement

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}]

Solution

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

Equation A:

(5b−2)2+4=25b2−20b(5b-2)^2 + 4 = 25b^2 - 20b

First, let's expand (5b−2)2(5b-2)^2: (5b−2)2=25b2−20b+4(5b-2)^2 = 25b^2 - 20b + 4

Now, substitute this back into the equation: 25b2−20b+4+4=25b2−20b25b^2 - 20b + 4 + 4 = 25b^2 - 20b 25b2−20b+8=25b2−20b25b^2 - 20b + 8 = 25b^2 - 20b

Clearly, 25b2−20b+8≠25b2−20b25b^2 - 20b + 8 \neq 25b^2 - 20b, so Equation A is not a true identity.

Equation B:

(2x2+y2)(2x2−y2)=4x4−y4(2x^2 + y^2)(2x^2 - y^2) = 4x^4 - y^4

This is a difference of squares on the left-hand side: (2x2+y2)(2x2−y2)=(2x2)2−(y2)2(2x^2 + y^2)(2x^2 - y^2) = (2x^2)^2 - (y^2)^2

Simplify: (2x2)2−(y2)2=4x4−y4(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(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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Identities
Difference of Squares

Formulas

(a - b)^2 = a^2 - 2ab + b^2
(a^2 - b^2) = (a + b)(a - b)

Theorems

Difference of Squares Theorem

Suitable Grade Level

Grades 9-10

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