What is the Identity (a-b)^2?

S6-SA1-0042

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity (a-b)^2 is a fundamental algebraic formula that helps us expand or simplify expressions where a difference between two terms is squared. It states that when you square the difference between two numbers or variables, the result is the square of the first term, minus two times the product of the two terms, plus the square of the second term.

Simple Example

Quick Example

Imagine you have a cricket ground that is 100 meters long. If you want to make it 2 meters shorter on each side for a practice match, the new length would be (100 - 2) meters. Squaring this to find an area (if it were a square) would be (100 - 2)^2. This identity helps calculate such values easily without first doing the subtraction.

Worked Example

Step-by-Step

Let's expand (5x - 3)^2 using the identity (a-b)^2 = a^2 - 2ab + b^2.

Step 1: Identify 'a' and 'b'. Here, a = 5x and b = 3.
---Step 2: Substitute 'a' and 'b' into the formula a^2 - 2ab + b^2.
So, (5x)^2 - 2(5x)(3) + (3)^2.
---Step 3: Calculate the first term, (5x)^2.
(5x)^2 = 5^2 * x^2 = 25x^2.
---Step 4: Calculate the middle term, -2(5x)(3).
-2 * 5 * x * 3 = -30x.
---Step 5: Calculate the last term, (3)^2.
(3)^2 = 9.
---Step 6: Combine all the terms.
25x^2 - 30x + 9.

Answer: (5x - 3)^2 = 25x^2 - 30x + 9

Why It Matters

This identity is crucial in many fields, from designing efficient algorithms in AI/ML to calculating trajectories in Space Technology. Engineers use it to simplify complex equations, and physicists apply it in various formulas. Understanding it can open doors to careers in software development, scientific research, and engineering.

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 negative sign in the middle term, writing a^2 + 2ab + b^2 | CORRECTION: The identity is for (a-b)^2, so the middle term is always negative: -2ab.

MISTAKE: Incorrectly squaring terms with coefficients, e.g., (3x)^2 = 3x^2 | CORRECTION: Square both the coefficient and the variable: (3x)^2 = 3^2 * x^2 = 9x^2.

Practice Questions

Try It Yourself

QUESTION: Expand (y - 7)^2. | ANSWER: y^2 - 14y + 49

QUESTION: If a = 4p and b = 5, what is (a - b)^2? | ANSWER: 16p^2 - 40p + 25

QUESTION: Simplify (100 - 1)^2 using the identity. | ANSWER: 9801

MCQ

Quick Quiz

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

4m^2 - 16

4m^2 + 16m + 16

4m^2 - 16m + 16

2m^2 - 8m + 16

The Correct Answer Is:

Using the identity (a-b)^2 = a^2 - 2ab + b^2, with a=2m and b=4, we get (2m)^2 - 2(2m)(4) + (4)^2 = 4m^2 - 16m + 16.

Real World Connection

In the Real World

This identity is used in computer graphics when scaling objects or calculating distances between points, like when your phone's GPS (Global Positioning System) app calculates how far you are from a destination. It also helps in optimizing code for faster processing in apps like UPI, where quick calculations are essential for secure transactions.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of the variables involved. | EXPANSION: Writing an expression as a sum or difference of terms. | TERM: A single number or variable, or numbers and variables multiplied together. | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

What's Next

What to Learn Next

Great job mastering (a-b)^2! Next, you should explore the identity (a+b)^2. It's very similar but involves addition, and understanding both will give you a strong foundation for more advanced algebraic manipulations.