Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem Statement**
1) Prove the following identity: \((x + y)^2 - (x - y)^2 = 4xy\).
**Explanation**
The problem requires proving the algebraic identity by expanding both squared terms on the left-hand side and simplifying the expression to show it equals the right-hand side, \(4xy\).
**Solution Steps**
1. Expand \((x + y)^2\):
\[
(x + y)^2 = x^2 + 2xy + y^2
\]
2. Expand \((x - y)^2\):
\[
(x - y)^2 = x^2 - 2xy + y^2
\]
3. Substitute the expanded forms into the identity:
\[
(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)
\]
4. Simplify the expression:
\[
x^2 + 2xy + y^2 - x^2 + 2xy - y^2
\]
5. Cancel out like terms:
\[
(x^2 - x^2) + (2xy + 2xy) + (y^2 - y^2) = 4xy
\]
**Conclusion**
The left-hand side simplifies to \(4xy\), proving the identity as required:
\[
(x + y)^2 - (x - y)^2 = 4xy
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47ebf8e0-0639-4c02-a098-1722152a8587%2F89cb92b9-dc05-4922-ac7e-cebc29a1d7d1%2Fzdub27_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
1) Prove the following identity: \((x + y)^2 - (x - y)^2 = 4xy\).
**Explanation**
The problem requires proving the algebraic identity by expanding both squared terms on the left-hand side and simplifying the expression to show it equals the right-hand side, \(4xy\).
**Solution Steps**
1. Expand \((x + y)^2\):
\[
(x + y)^2 = x^2 + 2xy + y^2
\]
2. Expand \((x - y)^2\):
\[
(x - y)^2 = x^2 - 2xy + y^2
\]
3. Substitute the expanded forms into the identity:
\[
(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)
\]
4. Simplify the expression:
\[
x^2 + 2xy + y^2 - x^2 + 2xy - y^2
\]
5. Cancel out like terms:
\[
(x^2 - x^2) + (2xy + 2xy) + (y^2 - y^2) = 4xy
\]
**Conclusion**
The left-hand side simplifies to \(4xy\), proving the identity as required:
\[
(x + y)^2 - (x - y)^2 = 4xy
\]
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