Factorising by splitting the middle term: Class 9 practice
To factorise ax² + bx + c, find two numbers whose product is ac and sum is b. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Factors of integers
- Taking out common factors
The idea behind factorising by splitting the middle term
To factorise ax² + bx + c, find two numbers whose product is ac and sum is b. Split bx into those two terms and group.
Step-by-step method
- Compute ac.
- Find two numbers with product ac and sum b.
- Split the middle term and group in pairs.
- Take out the common bracket and check by expanding.
Worked example 1 (Basic)
Factorise x² − 5x + 4.
- ac = 4; numbers -1 and -4 (product 4, sum -5)
- Group and factor: (x − 4)(x − 1)
- Answer: (x − 4)(x − 1)
Worked example 2 (Standard)
Factorise x² + 9x + 8.
- ac = 8; numbers 8 and 1 (product 8, sum 9)
- Group and factor: (x + 1)(x + 8)
- Answer: (x + 1)(x + 8)
Worked example 3 (Challenge)
Factorise 2x² + 26x + 72.
- ac = 144; numbers 18 and 8 (product 144, sum 26)
- Group and factor: (2x + 8)(x + 9)
- Answer: (2x + 8)(x + 9)
Common mistakes and misconceptions
- Choosing numbers with the right sum but wrong product
- Sign errors with negative c
Practice questions
- (Basic) Factorise x² + 15x + 56.
- (Basic) Factorise 3x² + 15x − 18.
- (Standard) Factorise x² + 1x − 20.
- (Standard) Factorise 3x² + 1x − 2.
- (Challenge) Factorise 4x² − 1x − 14.
- (Challenge) Factorise 4x² − 20x + 24.
Answer key with full working
- Q1: ac = 56; numbers 8 and 7 (product 56, sum 15) → Group and factor: (x + 7)(x + 8) → Answer: (x + 7)(x + 8)
- Q2: ac = -54; numbers 18 and -3 (product -54, sum 15) → Group and factor: (3x − 3)(x + 6) → Answer: (3x − 3)(x + 6)
- Q3: ac = -20; numbers 5 and -4 (product -20, sum 1) → Group and factor: (x − 4)(x + 5) → Answer: (x − 4)(x + 5)
- Q4: ac = -6; numbers 3 and -2 (product -6, sum 1) → Group and factor: (3x − 2)(x + 1) → Answer: (3x − 2)(x + 1)
- Q5: ac = -56; numbers -8 and 7 (product -56, sum -1) → Group and factor: (4x + 7)(x − 2) → Answer: (4x + 7)(x − 2)
- Q6: ac = 96; numbers -12 and -8 (product 96, sum -20) → Group and factor: (4x − 8)(x − 3) → Answer: (4x − 8)(x − 3)
Syllabus note
This skill is listed under "Polynomials" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is factorising by splitting the middle term in?
"Polynomials", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.