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Splitting the Middle Term
A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with a != 1, split the middle using two numbers that add to b and multiply to ac. Then group the four terms. This is the everyday method for Class 9 trinomials that are not perfect squares.
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Splitting the Middle Term
What is it?
A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with a != 1, split the middle using two numbers that add to b and multiply to ac. Then group the four terms. This is the everyday method for Class 9 trinomials that are not perfect squares.
In simple words
- A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p.
- For x^2 + p x + q, hunt integers m, n with product q and sum p.
- For ax^2 + bx + c, compute ac.
- If no integer pair has product q (or ac) and sum p (or b), the trinomial may not factor nicely over the integers.
Important terms
product q: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p.
sum p: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p.
ax^2 + bx + c: For ax^2 + bx + c with a !
ac: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p.
product: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p.
What you should learn
- Explain Splitting The Middle Term in simple mathematical language.
- Apply product q and sum p correctly.
- Apply split ac correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
Examples
- x2 + 5x + 6 = (x + 2)(x + 3)
- x2 - 5x + 6 = (x - 2)(x - 3)
- x2 + x - 6 = (x + 3)(x - 2)
PRODUCT Q AND SUM P
For x^2 + p x + q, hunt integers m, n with product q and sum p. Then x^2 + p x + q = (x + m)(x + n). Signs follow the product: both plus if q and p are plus; both minus if q is plus and p is minus; mixed if q is minus.
SPLIT AC
For ax^2 + bx + c, compute ac. Find two numbers that multiply to ac and add to b. Rewrite bx as the sum of those two terms, then factor by grouping. Example: 2x^2 + 7x + 3 has ac = 6; 6 and 1 multiply to 6 and add to 7, so 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Recommended for Class 9
Full school-level explanation
A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with a != 1, split the middle using two numbers that add to b and multiply to ac. Then group the four terms. This is the everyday method for Class 9 trinomials that are not perfect squares.
Monic first, then the ac method:
WHAT — Core idea
A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with a != 1, split the middle using two numbers that add to b and multiply to ac. Then group the four terms. This is the everyday method for Class 9 trinomials that are not perfect squares.
WHY — Why it matters
If no integer pair has product q (or ac) and sum p (or b), the trinomial may not factor nicely over the integers. Still check a perfect-square test before you give up. Always expand to confirm.
HOW — Method and examples
1. PRODUCT Q AND SUM P: For x^2 + p x + q, hunt integers m, n with product q and sum p. 2. SPLIT AC: For ax^2 + bx + c, compute ac.
HOW — Answer in an exam
For a != 1, the product you split is ac, not c. Using product c is the usual slip. Write the two middle terms and group; do not jump to factors without checking the outer product a x last = ac.
PRODUCT Q AND SUM P
For x^2 + p x + q, hunt integers m, n with product q and sum p. Then x^2 + p x + q = (x + m)(x + n). Signs follow the product: both plus if q and p are plus; both minus if q is plus and p is minus; mixed if q is minus.
Remember
- product q
- sum p
Examples
- x2 + 5x + 6 = (x + 2)(x + 3)
- x2 - 5x + 6 = (x - 2)(x - 3)
- x2 + x - 6 = (x + 3)(x - 2)
SPLIT AC
For ax^2 + bx + c, compute ac. Find two numbers that multiply to ac and add to b. Rewrite bx as the sum of those two terms, then factor by grouping. Example: 2x^2 + 7x + 3 has ac = 6; 6 and 1 multiply to 6 and add to 7, so 2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Remember
- ax^2 + bx + c
- ac
Examples
- 2x2 + 7x + 3 = (2x + 1)(x + 3)
- 3x2 + 10x + 8 = (3x + 4)(x + 2)
- 6x2 - 5x - 6 = (3x + 2)(2x - 3)
Remember
- product q
- sum p
- ax^2 + bx + c
- ac
- product
- sum
- ac
- c
How the ideas connect
If no integer pair has product q (or ac) and sum p (or b), the trinomial may not factor nicely over the integers. Still check a perfect-square test before you give up. Always expand to confirm.
Deeper understanding
Go beyond memorising Splitting the Middle Term. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
If no integer pair has product q (or ac) and sum p (or b), the trinomial may not factor nicely over the integers. Still check a perfect-square test before you give up. Always expand to confirm.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- product q
- sum p
- ax^2 + bx + c
- ac
- product q: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need t
- sum p: A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need t
For a != 1, the product you split is ac, not c. Using product c is the usual slip. Write the two middle terms and group; do not jump to factors without checking the outer product a x last = ac.
For long answers on Splitting the Middle Term: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- product q
- sum p
- ax^2 + bx + c
- ac
Important formulas
- product q
- sum p
- ax^2 + bx + c
Common exam traps
- For a !
- = 1, the product you split is ac, not c.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
For a !
- Mistake
= 1, the product you split is ac, not c.
- Mistake
Using product c is the usual slip.
- Mistake
Write the two middle terms and group; do not jump to factors without checking the outer product a x last = ac.
- Mistake
For a != 1, the product you split is ac, not c. Using product c is the usual slip. Write the two middle terms and group; do not jump to factors without checking
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
product q
sum p
ax^2 + bx + c
ac
A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with
- Definition
- A quadratic trinomial x^2 + p x + q factors as (x + m)(x + n) when m + n = p and m n = q: you need two numbers with product q and sum p. For ax^2 + bx + c with
- Example
- x2 + 5x + 6 = (x + 2)(x + 3)
- Critical warning
- For a !
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Exam Practice
Define
Q1. Define Splitting the Middle Term.
Explain
Q2. Explain: PRODUCT Q AND SUM P.
Example
Q3. Give one clear example of Splitting the Middle Term and say why it fits.
Mistake
Q4. State one common mistake students make about Splitting the Middle Term.
Exam note
Q5. Write a short exam note on Splitting the Middle Term: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.