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What is the Factorisation of a Quadratic Trinomial?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation of a quadratic trinomial means breaking down a mathematical expression with three terms (a trinomial) and the highest power of the variable being 2 (quadratic) into a product of simpler expressions, usually two binomials. Think of it like finding the ingredients that multiply together to make a cake.
Simple Example
Quick Example
Imagine you have a rectangle with an area described by the expression x^2 + 5x + 6. Factorising this means finding the length and width of the rectangle. We'll find that the length is (x + 2) and the width is (x + 3), because (x + 2) multiplied by (x + 3) gives x^2 + 5x + 6.
Worked Example
Step-by-Step
Let's factorise the quadratic trinomial: x^2 + 7x + 10
1. Identify the constant term (10) and the coefficient of the middle term (7).
---2. Find two numbers that multiply to give 10 AND add up to give 7. Let's list factors of 10: (1, 10), (2, 5). Now, which pair adds to 7? (2 + 5 = 7). So, 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 in pairs: (x^2 + 2x) + (5x + 10).
---5. Factor out the common term from each pair. From (x^2 + 2x), 'x' is common, so it becomes x(x + 2). From (5x + 10), '5' is common, so it becomes 5(x + 2).
---6. Now we have x(x + 2) + 5(x + 2). Notice that (x + 2) is common to both terms.
---7. Factor out (x + 2): (x + 2)(x + 5).
Answer: The factorisation of x^2 + 7x + 10 is (x + 2)(x + 5).
Why It Matters
Factorisation helps us solve complex problems by breaking them into smaller, manageable parts. In computer science, it's used in cryptography to secure online transactions, like when you use UPI. Engineers use it to design structures and predict how things will behave, ensuring our bridges and buildings are safe.
Common Mistakes
MISTAKE: Students often forget to check if the two numbers they picked for the middle term also multiply to the constant term. | CORRECTION: Always verify both conditions: the two numbers must multiply to the constant term AND add up to the coefficient of the middle term.
MISTAKE: Incorrectly grouping terms or factoring out the wrong common factor, especially with negative signs. | CORRECTION: Be very careful with signs. Double-check your factoring by multiplying back to see if you get the original expression for each grouped pair.
MISTAKE: Not finding any common factor in the final step, meaning the two binomials in parentheses are different. | CORRECTION: If the expressions in the parentheses (like (x+2) in the example) are not identical after grouping, it means there's a mistake in the previous steps. Recheck your numbers and factoring.
Practice Questions
Try It Yourself
QUESTION: Factorise x^2 + 6x + 8 | ANSWER: (x + 2)(x + 4)
QUESTION: Factorise y^2 - 9y + 14 | ANSWER: (y - 2)(y - 7)
QUESTION: Factorise m^2 + m - 20 | ANSWER: (m + 5)(m - 4)
MCQ
Quick Quiz
Which of the following is the correct factorisation of x^2 + 8x + 15?
(x + 3)(x + 5)
(x + 2)(x + 6)
(x + 1)(x + 15)
(x - 3)(x - 5)
The Correct Answer Is:
A
For x^2 + 8x + 15, we need two numbers that multiply to 15 and add to 8. These numbers are 3 and 5. So, the correct factorisation is (x + 3)(x + 5).
Real World Connection
In the Real World
Imagine you're a data scientist analysing cricket match statistics. You might use factorisation to simplify complex equations that predict player performance or team scores. It helps break down big data problems into smaller, easier-to-understand parts, just like how companies like Dream11 use math to make their games fair and exciting.
Key Vocabulary
Key Terms
TRINOMIAL: An algebraic expression with three terms, like x^2 + 5x + 6 | QUADRATIC: An expression where the highest power of the variable is 2, like x^2 | FACTORISATION: Breaking down an expression into a product of simpler ones | BINOMIAL: An algebraic expression with two terms, like (x + 2)
What's Next
What to Learn Next
Great job learning about factorisation! Next, you can explore 'Solving Quadratic Equations by Factorisation'. This builds directly on what you've learned, showing you how to find the exact values of 'x' that make these expressions equal to zero.
