What is Factorisation of a Quadratic Trinomial?

S3-SA1-0828

Grade Level:

Class 6

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

Definition

What is it?

Factorisation of a quadratic trinomial is like breaking down a number into its prime factors. It means writing a mathematical expression (with x^2, x, and a constant) as a product of two simpler expressions, usually two binomials. This helps us find the 'roots' or solutions for when the expression equals zero.

Simple Example

Quick Example

Imagine you have a rectangle whose area is given by the expression x^2 + 5x + 6. Factorisation helps you find the length and breadth of this rectangle in terms of 'x'. The length might be (x+2) and the breadth might be (x+3), because (x+2) * (x+3) gives you x^2 + 5x + 6.

Worked Example

Step-by-Step

Let's factorise the quadratic trinomial: x^2 + 7x + 10

1. Identify the coefficients: The coefficient of x^2 is 1, the coefficient of x is 7, and the constant term is 10.
---2. Find two numbers that multiply to the constant term (10) and add up to the coefficient of x (7). Let's list factors of 10: (1, 10), (2, 5). Now check their sums: 1+10=11, 2+5=7. The numbers are 2 and 5.
---3. Rewrite the middle term (7x) using these two numbers: x^2 + 2x + 5x + 10.
---4. Group the terms: (x^2 + 2x) + (5x + 10).
---5. Factor out the common term from each group: x(x + 2) + 5(x + 2).
---6. Notice that (x + 2) is a common factor in both parts. Factor it out: (x + 2)(x + 5).
---Answer: The factorisation of x^2 + 7x + 10 is (x + 2)(x + 5).

Why It Matters

Understanding factorisation is crucial for solving many real-world problems. Engineers use it to design structures, physicists use it to model motion, and computer scientists use it in algorithms. It's a fundamental skill for careers in AI/ML, Data Science, and even game development.

Common Mistakes

MISTAKE: Not finding the correct pair of numbers that multiply to the constant AND add to the middle term's coefficient. | CORRECTION: Always list all factor pairs of the constant term and then check their sums to find the correct pair.

MISTAKE: Incorrectly grouping terms or making sign errors when factoring out common terms. For example, x^2 - 5x + 6, if you use -2 and -3, you might write x(x+2) - 3(x+2) instead of x(x-2) - 3(x-2). | CORRECTION: Double-check your signs carefully when grouping and factoring, especially when negative numbers are involved.

MISTAKE: Forgetting to check the factorisation by multiplying the binomials back. | CORRECTION: After factorising, always multiply the two binomials you found. If you get the original trinomial back, your factorisation is correct!

Practice Questions

Try It Yourself

QUESTION: Factorise: x^2 + 8x + 15 | ANSWER: (x + 3)(x + 5)

QUESTION: Factorise: y^2 - 10y + 24 | ANSWER: (y - 4)(y - 6)

QUESTION: Factorise: 2x^2 + 11x + 5 (Hint: Multiply the coefficient of x^2 by the constant term first!) | ANSWER: (2x + 1)(x + 5)

MCQ

Quick Quiz

Which of the following is the correct factorisation of x^2 + 6x + 8?

(x + 2)(x + 4)

(x + 1)(x + 8)

(x + 3)(x + 3)

(x + 2)(x + 6)

The Correct Answer Is:

For x^2 + 6x + 8, we need two numbers that multiply to 8 and add to 6. The numbers 2 and 4 satisfy this, as 2*4=8 and 2+4=6. So, (x+2)(x+4) is the correct factorisation.

Real World Connection

In the Real World

Imagine an architect designing a new metro station in Bengaluru. They might use quadratic equations to calculate the optimal curve for a roof or the stress on a beam. Factorisation helps them quickly find the dimensions or critical points for these designs, ensuring safety and efficiency.

Key Vocabulary

Key Terms

TRINOMIAL: A polynomial with three terms, like x^2 + 5x + 6 | QUADRATIC: Involving the second power of an unknown quantity, like x^2 | FACTORISATION: Breaking down an expression into a product of simpler expressions | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression, e.g., 5 in 5x | CONSTANT TERM: A term in an algebraic expression that has a value that is fixed and does not change, e.g., 6 in x^2 + 5x + 6

What's Next

What to Learn Next

Great job understanding factorisation! Next, you can learn about 'Solving Quadratic Equations by Factorisation'. This builds directly on what you've learned and shows you how to use factorisation to find specific values for 'x' that make an equation true.