What is Factorisation by Splitting the Middle Term?

S6-SA1-0053

Grade Level:

Class 10

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

Definition

What is it?

Factorisation by Splitting the Middle Term is a method used to break down a quadratic expression (like ax^2 + bx + c) into a product of two simpler linear expressions. It works by rewriting the middle term 'bx' as a sum of two terms, allowing us to group and factorise the expression.

Simple Example

Quick Example

Imagine you have a big pile of 12 laddoos, and you want to arrange them into equal rows. You could have 1 row of 12, 2 rows of 6, 3 rows of 4, and so on. Factorisation is like finding these 'rows' (factors). Splitting the middle term helps us find the right 'rows' for a more complex mathematical expression.

Worked Example

Step-by-Step

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

1. Identify a, b, and c: Here, a = 1, b = 7, c = 10.

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2. Find two numbers that multiply to 'ac' and add up to 'b'. Here, ac = 1 * 10 = 10 and b = 7. We need two numbers that multiply to 10 and add to 7. These numbers are 2 and 5 (since 2 * 5 = 10 and 2 + 5 = 7).

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3. Split the middle term 'bx' using these two numbers. Rewrite 7x as 2x + 5x. So, the expression becomes: x^2 + 2x + 5x + 10.

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4. Group the terms into two pairs: (x^2 + 2x) + (5x + 10).

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5. Factor out the common term from each pair. From (x^2 + 2x), common term is 'x', so it becomes x(x + 2). From (5x + 10), common term is '5', so it becomes 5(x + 2).

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6. Now the expression is: x(x + 2) + 5(x + 2).

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7. Notice that (x + 2) is common in both terms. Factor out (x + 2). This gives: (x + 2)(x + 5).

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ANSWER: The factors of x^2 + 7x + 10 are (x + 2) and (x + 5).

Why It Matters

This method is super important in engineering and science because it helps simplify complex equations, making them easier to solve. For instance, in Physics, it's used to understand projectile motion or electrical circuits. Future engineers and data scientists use this to build models and solve problems efficiently.

Common Mistakes

MISTAKE: Not finding the correct pair of numbers that multiply to 'ac' and add to 'b'. | CORRECTION: Always list all pairs of factors for 'ac' and then check which pair sums up to 'b'. Remember to consider negative numbers too!

MISTAKE: Incorrectly factoring out the common terms from the grouped pairs, especially with negative signs. | CORRECTION: Double-check your grouping and ensure the terms inside the parentheses are identical after factoring out. For example, if you have -3x - 6, factoring out -3 gives -3(x + 2), not -3(x - 2).

MISTAKE: Forgetting to factor out the common bracket in the final step. | CORRECTION: After grouping and factoring, you should always end up with a common binomial factor (like (x+2) in the example). If you don't, recheck your previous steps.

Practice Questions

Try It Yourself

QUESTION: Factorise: x^2 + 5x + 6 | ANSWER: (x + 2)(x + 3)

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

QUESTION: Factorise: 2x^2 + 11x + 5 | ANSWER: (2x + 1)(x + 5)

MCQ

Quick Quiz

Which of the following is a factor of x^2 + 9x + 14?

(x + 2)

(x + 5)

(x - 7)

(x - 2)

The Correct Answer Is:

For x^2 + 9x + 14, we need two numbers that multiply to 14 and add to 9. These numbers are 2 and 7. So, we split 9x into 2x + 7x, leading to (x+2)(x+7). Thus, (x+2) is a factor.

Real World Connection

In the Real World

This method is crucial in computer programming, especially when designing algorithms for tasks like image processing or financial modeling. For example, a software engineer might use factorisation to optimise calculations for predicting stock market trends or to efficiently process data from a large e-commerce platform like Flipkart or Amazon.

Key Vocabulary

Key Terms

Quadratic Expression: An expression of the form ax^2 + bx + c, where 'a' is not zero. | Factorisation: Breaking down an expression into a product of simpler expressions. | Middle Term: The 'bx' term in a quadratic expression ax^2 + bx + c. | Binomial: An algebraic expression with two terms, like (x + 5).

What's Next

What to Learn Next

Great job mastering this! Next, you should explore 'Factorisation of Polynomials by the Factor Theorem'. It builds on understanding factors and will help you factorise even more complex expressions beyond quadratics, which is useful in higher mathematics.