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What is the Factorisation of Difference of Two Squares?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation of Difference of Two Squares is a special algebraic identity that helps us break down an expression where one perfect square is subtracted from another perfect square. It states that a^2 - b^2 can always be written as (a - b)(a + b). This identity makes complex calculations simpler.
Simple Example
Quick Example
Imagine you have two square plots of land. One is 10 meters by 10 meters (area 100 sq m) and another is 7 meters by 7 meters (area 49 sq m). If you want to find the difference in their areas, you can do 100 - 49. Using our formula, this is 10^2 - 7^2 = (10 - 7)(10 + 7) = 3 * 17 = 51 sq m. Much faster!
Worked Example
Step-by-Step
Let's factorise the expression 25x^2 - 49y^2.
Step 1: Identify if both terms are perfect squares. Here, 25x^2 is (5x)^2 and 49y^2 is (7y)^2.
---Step 2: Recognize the form a^2 - b^2. In our case, a = 5x and b = 7y.
---Step 3: Apply the identity: a^2 - b^2 = (a - b)(a + b).
---Step 4: Substitute the values of 'a' and 'b' into the identity.
---Step 5: So, 25x^2 - 49y^2 = (5x - 7y)(5x + 7y).
---Answer: (5x - 7y)(5x + 7y)
Why It Matters
This concept is super important for simplifying complex math problems, especially in higher classes! Engineers use it to design structures, and computer scientists use it in creating secure online systems (cryptography). It helps in fields like Data Science and AI to optimise calculations.
Common Mistakes
MISTAKE: Students sometimes factorise a^2 + b^2 as (a + b)(a - b). | CORRECTION: The identity only works for a DIFFERENCE (subtraction) of two squares, not a sum. a^2 + b^2 cannot be factored this way using real numbers.
MISTAKE: For 4x^2 - 9, students might incorrectly write (4x - 3)(4x + 3). | CORRECTION: Remember to take the square root of the ENTIRE first term. The square root of 4x^2 is 2x, not 4x. So it should be (2x - 3)(2x + 3).
MISTAKE: For x^2 - 5, students try to factorise it directly. | CORRECTION: The number 5 is not a perfect square. While it can be written as (sqrt(5))^2, for basic factorisation, ensure both terms are perfect squares of integers or simple algebraic expressions.
Practice Questions
Try It Yourself
QUESTION: Factorise x^2 - 16. | ANSWER: (x - 4)(x + 4)
QUESTION: Factorise 81 - 64y^2. | ANSWER: (9 - 8y)(9 + 8y)
QUESTION: Factorise 100m^2 - 1. | ANSWER: (10m - 1)(10m + 1)
MCQ
Quick Quiz
Which of the following is the correct factorisation of 36p^2 - 25?
(6p - 5)(6p - 5)
(6p + 5)(6p + 5)
(6p - 5)(6p + 5)
(36p - 25)(36p + 25)
The Correct Answer Is:
C
The expression 36p^2 - 25 is a difference of two squares, where a = 6p (since (6p)^2 = 36p^2) and b = 5 (since 5^2 = 25). So, using the identity a^2 - b^2 = (a - b)(a + b), we get (6p - 5)(6p + 5).
Real World Connection
In the Real World
This identity is used in computer graphics to quickly calculate distances and optimize rendering. For example, when designing video games or animations, developers use similar mathematical shortcuts to make characters move smoothly and environments load faster, even on your mobile phone!
Key Vocabulary
Key Terms
FACTORISATION: Breaking down an expression into a product of simpler expressions. | IDENTITY: An equation that is true for all possible values of the variables. | PERFECT SQUARE: A number or expression that is the square of an integer or another expression (e.g., 9 is a perfect square because 3^2 = 9). | DIFFERENCE: The result of subtracting one number or expression from another.
What's Next
What to Learn Next
Great job understanding this! Next, you should learn about factorisation of quadratic expressions (like x^2 + 5x + 6). This will build on what you've learned here and open up even more ways to solve algebraic puzzles.
