What is (a-b) Whole Square?

S3-SA1-0042

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

(a-b) whole square, also written as (a-b)^2, means multiplying the expression (a-b) by itself. It expands to a^2 - 2ab + b^2. This is a fundamental algebraic identity used to quickly solve problems involving the difference of two terms squared.

Simple Example

Quick Example

Imagine you have Rs. 50 (a) and you spend Rs. 5 (b) on a packet of biscuits. You want to find out what happens if you square the remaining amount. Instead of (50-5)^2 = 45^2 = 2025, you can use the formula: 50^2 - 2*50*5 + 5^2. This gives 2500 - 500 + 25 = 2025. Both methods give the same answer!

Worked Example

Step-by-Step

Let's calculate (10 - 3)^2 using the identity (a-b)^2 = a^2 - 2ab + b^2.
---Step 1: Identify 'a' and 'b'. Here, a = 10 and b = 3.
---Step 2: Substitute 'a' and 'b' into the formula: (10)^2 - 2*(10)*(3) + (3)^2.
---Step 3: Calculate the square of 'a': 10^2 = 100.
---Step 4: Calculate the product 2ab: 2 * 10 * 3 = 60.
---Step 5: Calculate the square of 'b': 3^2 = 9.
---Step 6: Combine the results: 100 - 60 + 9.
---Step 7: Perform the subtraction and addition: 40 + 9 = 49.
---Answer: So, (10 - 3)^2 = 49.

Why It Matters

This identity helps simplify complex calculations in fields like engineering and physics, making problems easier to solve. In data science, it's used in algorithms to measure differences, and in computer science, it helps optimize code for faster processing. Understanding this helps you build a strong foundation for future careers in technology and research.

Common Mistakes

MISTAKE: Students often write (a-b)^2 as a^2 - b^2. | CORRECTION: Remember the middle term! (a-b)^2 is a^2 - 2ab + b^2, not a^2 - b^2. The term -2ab is crucial.

MISTAKE: Forgetting the minus sign for the middle term, writing a^2 + 2ab + b^2. | CORRECTION: The identity for (a-b)^2 has a minus sign before the 2ab term. (a+b)^2 has a plus sign.

MISTAKE: Applying the square only to 'a' or 'b' individually, like a^2 - b. | CORRECTION: The 'whole square' means the entire expression (a-b) is multiplied by itself, leading to three terms in the expansion.

Practice Questions

Try It Yourself

QUESTION: Expand (x - 5)^2. | ANSWER: x^2 - 10x + 25

QUESTION: If a = 7 and b = 2, calculate (a - b)^2 using the identity. | ANSWER: (7-2)^2 = 5^2 = 25. Using identity: 7^2 - 2*7*2 + 2^2 = 49 - 28 + 4 = 21 + 4 = 25.

QUESTION: A square field has a side length of (2p - 3) meters. What is the area of the field? | ANSWER: Area = (side)^2 = (2p - 3)^2 = (2p)^2 - 2*(2p)*(3) + (3)^2 = 4p^2 - 12p + 9 square meters.

MCQ

Quick Quiz

Which of the following is the correct expansion of (4 - y)^2?

16 - y^2

16 - 4y + y^2

16 - 8y + y^2

16 + 8y + y^2

The Correct Answer Is:

Using the identity (a-b)^2 = a^2 - 2ab + b^2, where a=4 and b=y. So, 4^2 - 2*4*y + y^2 = 16 - 8y + y^2. Options A, B, and D miss terms or have incorrect signs.

Real World Connection

In the Real World

Imagine an architect designing a building. If they need to reduce the side of a square room by a certain amount (say, from 'x' meters to 'x-2' meters), they can quickly calculate the new area using (x-2)^2. This helps them plan space and materials efficiently, saving time and resources on construction projects across India.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of its variables. | EXPANSION: Writing an expression as a sum or difference of terms. | SQUARE: Multiplying a number or expression by itself. | TERM: A single number or variable, or numbers and variables multiplied together.

What's Next

What to Learn Next

Great job understanding (a-b) whole square! Next, you should explore (a+b) whole square and then move on to the difference of squares identity, (a^2 - b^2). These identities are like powerful shortcuts that will make solving algebraic problems much faster and easier!