In-Depth: Algebraic Identities
CLAT Application & Relevance
Importance: Low. While explicitly listed as "Algebraic Identities - (a+b)², (a-b)², a² - b², etc.", direct questions on complex identities are very rare. Their primary relevance in CLAT QT is to aid in simplifying calculations, especially within complex caselets or Data Interpretation questions where numbers might appear in a form that can be quickly simplified using these identities. Focus on understanding and quickly applying the fundamental identities.
How it's tested: Indirectly, as a tool to simplify expressions or numbers encountered in a passage-based quantitative problem, rather than standalone algebra questions.
Section 1: Core Concepts & Important Identities
An algebraic identity is an equation that is true for all possible values of its variables. These identities are useful for simplifying algebraic expressions and performing quick calculations.
Fundamental Identities
- Square of a Binomial:
(a + b)² = a² + 2ab + b²(a - b)² = a² - 2ab + b²
- Difference of Squares:
a² - b² = (a - b)(a + b)
- Cube of a Binomial:
(a + b)³ = a³ + b³ + 3ab(a + b) = a³ + b³ + 3a²b + 3ab²(a - b)³ = a³ - b³ - 3ab(a - b) = a³ - b³ - 3a²b + 3ab²
- Sum/Difference of Cubes:
a³ + b³ = (a + b)(a² - ab + b²)a³ - b³ = (a - b)(a² + ab + b²)
- Square of a Trinomial:
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Section 2: Solved CLAT-Style Examples (Application in Simplification)
Example 1: Using Difference of Squares for Quick Calculation
Passage Context: "A financial report mentions that a particular asset's valuation was calculated based on the difference between the square of two large numbers: (195)² and (105)²."
Question: "What is the numerical value of (195)² - (105)²?"
Detailed Solution:
1. Recognize the Identity: This is in the form a² - b².
2. Apply the Formula: a² - b² = (a - b)(a + b)
Here, a = 195, b = 105.
(195)² - (105)² = (195 - 105)(195 + 105)
3. Simplify and Calculate:
= (90)(300)
= 27000.
Answer: The numerical value is 27,000. (Direct squaring and subtracting would be much slower).
Example 2: Applying Square of a Binomial (Caselet with Rates)
Passage Context: "A new legal process increased efficiency. Initially, a task took 'X' minutes. With the new process, it now takes (100 - 1)² minutes. If a similar task, when made simpler, took (100 + 1)² minutes. What is the difference in time between the simplified task and the new process task?"
Question: "What is the value of (100 + 1)² - (100 - 1)²?"
Detailed Solution:
1. Recognize the Pattern: This can be viewed as A² - B² where A = (100 + 1) and B = (100 - 1).
2. Apply Difference of Squares: A² - B² = (A - B)(A + B)
= [(100 + 1) - (100 - 1)] * [(100 + 1) + (100 - 1)]
= [100 + 1 - 100 + 1] * [100 + 1 + 100 - 1]
= [2] * [200]
= 400.
Alternatively (expanding binomials):
(100 + 1)² = 100² + 2*100*1 + 1² = 10000 + 200 + 1 = 10201
(100 - 1)² = 100² - 2*100*1 + 1² = 10000 - 200 + 1 = 9801
10201 - 9801 = 400.
Answer: The difference in time is 400 minutes.