Math
Algebra - Identity
MCQ

Introduction

Algebra is not just about solving for 'x'; it is about finding patterns. The most commonly used formula is (a + b)2 = a2 + 2ab + b2. This identity is used to quickly expand the square of a binomial and to calculate squares of large numbers without tedious multiplication. It bridges the gap between basic arithmetic and advanced calculus, saving time in complex calculations.

Explanation Step by Step

Think of an identity like a recipe. No matter what numbers you use, the result follows the same rule. For example, squaring 103 can be split into (100 + 3)2. Instead of long multiplication, we follow the a2 + 2ab + b2 pattern.

Sub-topics

Identity

An algebraic identity is an equality that holds true regardless of the values assigned to its variables. Key formula: (a + b)2 = a2 + 2ab + b2.

Examples

Example 1: Expand (x + 4)2.
Step 1: Identify a = x and b = 4.
Step 2: Apply formula a2 + 2ab + b2.
Step 3: x2 + 2(x)(4) + 42.
Answer: x2 + 8x + 16
Tricky Example (Real-Life): You want to fence a square garden with side length 99 ft. What is the area?
Step 1: Write 99 as (100 - 1).
Step 2: Use (a-b)2 = a2 - 2ab + b2.
Step 3: 10000 - 200 + 1.
Answer: 9801 sq. ft.

Tricks and Shortcuts

1. **The Base 100 Trick:** For numbers near 100, always use (100 + x) or (100 - x).
2. **Mental Math:** Visualize the "2ab" part as doubling the product of the two terms before adding the squares.

Common Mistakes

1. **Missing Middle Term:** Writing (a+b)2 = a2 + b2. Never forget the "2ab"!
2. **Squaring Negatives:** Thinking (-b)2 is negative. A square is always positive.

Practice Questions

Easy Questions

  1. Expand (y + 5)2.
  2. Evaluate (52)2 using (a+b)2.
  3. Tricky Question: If a = 1 and b = -1, what is the value of (a+b)2?

Medium Questions

  1. Find the value of (3x - 2y)2.
  2. Calculate (998)2 using a suitable identity.
  3. Tricky Question: A square picture frame has a side of (x - 3) units. Find the area expression.

Hard Questions

  1. If x2 + y2 = 29 and xy = 10, find the value of (x + y).
  2. Tricky Question: Solve for (10.1)2 using (a+b)2.
  3. Find the area of a square whose perimeter is 4x + 28.

Revision Summary

Identities like (a+b)2 = a2 + 2ab + b2 are universal tools. They help in mental calculation, geometry, and higher-level physics. Always include the 2ab term to ensure accuracy.

Which of the following is the expansion of (a - b + c)²?
A
a² + b² + c² - 2ab - 2bc + 2ca
B
a² + b² + c² - 2ab + 2bc - 2ca
C
a² + b² + c² + 2ab - 2bc + 2ca
D
a² + b² + c² - 2ab + 2ca
Explanation
According to the constant equation, considering the signs, since the terms of b are negative, the terms where b occurs will be negative. (2ab and 2bc will be subtracted).
Correct 3 Wrong 1