What is (a-b) Squared?

S3-SA1-0750

Grade Level:

Class 6

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

Definition

What is it?

(a-b) Squared, written as (a-b)^2, means multiplying the expression (a-b) by itself. It's a special algebraic identity that expands to a^2 - 2ab + b^2. This formula helps us quickly calculate the square of a difference between two numbers or terms.

Simple Example

Quick Example

Imagine you have 100 rupees (a) and you spend 2 rupees (b). You want to find (100-2) squared. Instead of first calculating 98 x 98, we can use the formula: (100)^2 - 2*(100)*(2) + (2)^2. This makes the calculation faster and easier!

Worked Example

Step-by-Step

Let's calculate (5 - 3)^2 using the formula.
---Step 1: Identify 'a' and 'b'. Here, a = 5 and b = 3.
---Step 2: Recall the formula: (a-b)^2 = a^2 - 2ab + b^2.
---Step 3: Substitute the values of 'a' and 'b' into the formula. So, (5-3)^2 = (5)^2 - 2*(5)*(3) + (3)^2.
---Step 4: Calculate the squares: (5)^2 = 25 and (3)^2 = 9.
---Step 5: Calculate the middle term: 2*(5)*(3) = 2*15 = 30.
---Step 6: Put it all together: 25 - 30 + 9.
---Step 7: Perform the subtraction and addition: 25 - 30 = -5, then -5 + 9 = 4.
---Step 8: So, (5-3)^2 = 4. (Check: 5-3 = 2, and 2^2 = 4. The formula works!)

Why It Matters

Understanding (a-b)^2 is crucial for solving complex problems in fields like Physics and Engineering, where you often need to simplify expressions quickly. Even in Computer Science, optimizing code can involve using such algebraic identities. It's a fundamental tool that helps engineers design bridges and scientists model weather patterns!

Common Mistakes

MISTAKE: Writing (a-b)^2 as a^2 - b^2 | CORRECTION: Remember the middle term! The correct expansion is a^2 - 2ab + b^2.

MISTAKE: Forgetting the minus sign in the middle term, writing a^2 + 2ab + b^2 | CORRECTION: For (a-b)^2, the middle term is always -2ab. The plus sign is for (a+b)^2.

MISTAKE: Only squaring 'a' and 'b' and ignoring the whole expression. For example, (7-2)^2 becomes 7^2 - 2^2 | CORRECTION: The entire difference (a-b) is multiplied by itself, not just the individual terms squared and subtracted.

Practice Questions

Try It Yourself

QUESTION: Expand (x-4)^2. | ANSWER: x^2 - 8x + 16

QUESTION: Calculate (98)^2 using the identity (100-2)^2. | ANSWER: (100-2)^2 = 100^2 - 2*100*2 + 2^2 = 10000 - 400 + 4 = 9604

QUESTION: If a = 12 and b = 7, find the value of (a-b)^2 using the formula. | ANSWER: (12-7)^2 = 12^2 - 2*12*7 + 7^2 = 144 - 168 + 49 = 25

MCQ

Quick Quiz

Which of the following is the correct expansion of (P-Q)^2?

P^2 + Q^2

P^2 - Q^2

P^2 - 2PQ + Q^2

P^2 + 2PQ + Q^2

The Correct Answer Is:

The correct identity for (a-b)^2 is a^2 - 2ab + b^2. Therefore, for (P-Q)^2, it expands to P^2 - 2PQ + Q^2. Options A and B are incorrect as they miss the middle term, and Option D is for (P+Q)^2.

Real World Connection

In the Real World

Imagine an engineer designing a mobile phone screen. If the screen's original length is 'L' and they need to reduce it by a small amount 'x' to fit a new frame, they might need to calculate the new area which involves (L-x)^2. This concept helps in precise measurements and design adjustments in manufacturing.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of its variables | EXPAND: To write an algebraic expression without parentheses by multiplying terms | SQUARE: To multiply 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)^2! Next, you should explore '(a+b) Squared' and then '(a+b)(a-b)'. These are also important algebraic identities that will make solving more complex math problems much easier and faster!