Middle Term Splitting Factorization: Examples + Worksheet

Factorization is a cornerstone concept in algebra, and among its many methods, Middle Term Splitting Factorization stands out for its simplicity and logic. This technique is primarily used to factor quadratic trinomials of the form ax² + bx + c, especially when the coefficient of x² is greater than 1.

Factorization dates back to ancient mathematicians in India and the Middle East. Indian mathematician Brahmagupta (598–668 CE) was among the first to describe methods to solve quadratic equations, laying the groundwork for what would later be refined into mid-term splitting in modern algebraic contexts. Today, mid-term splitting is widely taught in IGCSE Mathematics as a primary method for solving and factoring non-monic quadratic expressions.

Let’s dive into the method, learn the logic, and walk through progressively challenging examples before giving you a free downloadable worksheet with 20 practice problems and answers.

📘 What is Middle Term Splitting Factorization?

Mid-term splitting (also known as the method of decomposition) involves

Multiplying the first and last coefficient: a × c
Finding two numbers that multiply to a×c and add up to b, the middle term.
Rewrite the middle term using these two numbers.
Grouping and factoring by common terms.
Extracting the final binomial factors.

✅ 7 Solved Examples (Beginner to Advanced)

Example 1: x² + 5x + 6

This is a monic quadratic (coefficient of x² is 1).

Step 1: Multiply 1×6 = 6
Step 2: Find two numbers that multiply to 6 and add to 5 → 2 and 3
Step 3: x² + 2x + 3x + 6
Step 4: Group → (x² + 2x) + (3x + 6)
Step 5: x(x + 2) + 3(x + 2)
Final Answer: (x + 2)(x + 3)

Example 2: x² – 7x + 12

Step 1: 1×12 = 12
Step 2: -3 and -4 → (-3)(-4) = 12 and -3 + -4 = -7
Step 3: x² – 3x – 4x + 12
Step 4: (x² – 3x) – (4x – 12)
Step 5: x(x – 3) – 4(x – 3)
Final Answer: (x – 3)(x – 4)

Example 3: 2x² + 7x + 3

Step 1: 2×3 = 6
Step 2: 6 = 6×1, 6+1 = 7 → numbers are 6 and 1
Step 3: 2x² + 6x + x + 3
Step 4: (2x² + 6x) + (x + 3)
Step 5: 2x(x + 3) + 1(x + 3)
Final Answer: (2x + 1)(x + 3)

Example 4: 3x² – 11x – 4

Step 1: 3×-4 = -12
Step 2: Numbers = -12 and 1 (because -12 + 1 = -11)
Step 3: 3x² – 12x + x – 4
Step 4: (3x² – 12x) + (x – 4)
Step 5: 3x(x – 4) + 1(x – 4)
Final Answer: (3x + 1)(x – 4)

Example 5: 6x² + 5x – 6

Step 1: 6×-6 = -36
Step 2: 9 and -4 → 9 – 4 = 5
Step 3: 6x² + 9x – 4x – 6
Step 4: (6x² + 9x) – (4x + 6)
Step 5: 3x(2x + 3) – 2(2x + 3)
Final Answer: (3x – 2)(2x + 3)

Example 6: 4x² – 8x – 5

Step 1: 4×-5 = -20
Step 2: -10 and 2 → -10 + 2 = -8
Step 3: 4x² – 10x + 2x – 5
Step 4: (4x² – 10x) + (2x – 5)
Step 5: 2x(2x – 5) + 1(2x – 5)
Final Answer: (2x + 1)(2x – 5)

Example 7: 5x² + 13x – 6

Step 1: 5×-6 = -30
Step 2: 15 and -2 → 15 – 2 = 13
Step 3: 5x² + 15x – 2x – 6
Step 4: (5x² + 15x) – (2x + 6)
Step 5: 5x(x + 3) – 2(x + 3)
Final Answer: (5x – 2)(x + 3)

📄 Download Worksheet: Middle Term Splitting Factorization (20 Questions)

We’ve created a practice worksheet for mid-term splitting factorization with 20 problems from basic to advanced level. You’ll find mixed question types — positive, negative, and large coefficients — all aimed at improving your skills.

✅ Features:

Printable PDF
Online answers included
Great for revision or homework

👉 [Download Worksheet PDF]

✨ Conclusion: Master Mid Term Breaking Factorization with Confidence!

Mastering middle term splitting factorization is like unlocking a universal key to many algebraic problems in IGCSE Mathematics. As you’ve seen in the examples above, it is logical, procedural, and incredibly effective.

But solving these by yourself can feel tricky at times — especially when expressions look complicated. That’s why My Maths Club offers live online classes with expert guidance, tailored practice, and free downloadable resources.

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