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What is the Factorisation of a Perfect Square Trinomial?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation of a perfect square trinomial means breaking down a special type of three-term algebraic expression into the product of two identical binomials. These trinomials come from squaring a binomial, like (a + b)^2 or (a - b)^2. It's like finding the original building blocks of a number.
Simple Example
Quick Example
Imagine you have a square plot of land. If one side is (x + 3) meters, its area would be (x + 3) * (x + 3) or (x + 3)^2. When you expand this, you get x^2 + 6x + 9. Factorising x^2 + 6x + 9 means finding those original side lengths, which are (x + 3) and (x + 3).
Worked Example
Step-by-Step
Let's factorise the perfect square trinomial: 4x^2 + 12x + 9
---1. Identify the first and last terms. Check if they are perfect squares. Here, 4x^2 is (2x)^2 and 9 is (3)^2.
---2. Identify the square roots of the first and last terms. These are 2x and 3.
---3. Check if the middle term is twice the product of these square roots. The middle term is 12x. Is 2 * (2x) * (3) equal to 12x? Yes, 2 * 2x * 3 = 12x.
---4. Since all conditions are met and the middle term is positive, this is a perfect square trinomial of the form (a + b)^2. Here, a = 2x and b = 3.
---5. Write the factored form as (a + b)^2. So, (2x + 3)^2.
Answer: (2x + 3)^2 or (2x + 3)(2x + 3)
Why It Matters
Understanding factorisation helps you simplify complex equations in physics and engineering, making calculations easier. It's also crucial in computer science for optimising algorithms and in data science for pattern recognition. Future innovators in AI/ML and cryptography use these concepts to build secure systems and smart technologies.
Common Mistakes
MISTAKE: Not checking if the middle term matches 2ab | CORRECTION: Always verify that the middle term of the trinomial is exactly 2 times the product of the square roots of the first and last terms. For x^2 + 5x + 9, it's not a perfect square trinomial because 2 * x * 3 = 6x, not 5x.
MISTAKE: Confusing (a + b)^2 and (a - b)^2 forms | CORRECTION: If the middle term is positive, use (a + b)^2. If the middle term is negative, use (a - b)^2. For example, x^2 - 10x + 25 factors to (x - 5)^2, not (x + 5)^2.
MISTAKE: Forgetting to take the square root of the coefficient of x^2 | CORRECTION: If the first term is 9x^2, its square root is 3x, not just x. So, for 9x^2 + 24x + 16, the 'a' term is 3x, not x.
Practice Questions
Try It Yourself
QUESTION: Factorise y^2 + 14y + 49 | ANSWER: (y + 7)^2
QUESTION: Factorise 25p^2 - 20p + 4 | ANSWER: (5p - 2)^2
QUESTION: Factorise 36m^2 + 60mn + 25n^2 | ANSWER: (6m + 5n)^2
MCQ
Quick Quiz
Which of the following is the factorisation of x^2 - 16x + 64?
(x + 8)^2
(x - 8)^2
(x - 4)(x - 16)
(x + 4)(x + 16)
The Correct Answer Is:
B
The square root of x^2 is x, and the square root of 64 is 8. Since the middle term (-16x) is negative and equals 2 * x * (-8), the correct form is (x - 8)^2.
Real World Connection
In the Real World
Engineers designing bridges or buildings use algebraic expressions, including perfect square trinomials, to calculate areas, volumes, and structural strengths. For example, when calculating the surface area of a complex part in an ISRO rocket or optimising the design of a metro station, simplifying these expressions helps ensure safety and efficiency.
Key Vocabulary
Key Terms
TRINOMIAL: An algebraic expression with three terms, like x^2 + 2x + 1 | PERFECT SQUARE: A number or expression that is the square of another number or expression, like 9 (which is 3^2) or (x + 1)^2 | FACTORISATION: Breaking down an expression into a product of simpler expressions | BINOMIAL: An algebraic expression with two terms, like (x + 5)
What's Next
What to Learn Next
Great job mastering perfect square trinomials! Next, you should explore factorisation of general quadratic trinomials (like x^2 + bx + c) where the first and last terms might not be perfect squares. This will give you even more tools to solve complex algebraic problems.
