MathematicsChapter 4: Factorization and Algebraic Manipulation
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.
Learning objectives
- 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.
- Recognise and correct common mistakes.
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 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.
Formula
n = p and m n
Where / symbols
- 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.
- sum
- 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.
- c
- 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.
Overview
Monic first, then the ac method:
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.
Examples x^2 + 5x + 6 = (x + 2)(x + 3) · x^2 - 5x + 6 = (x - 2)(x - 3) · x^2 + 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 2x2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Examples 2x^2 + 7x + 3 = (2x + 1)(x + 3) · 3x^2 + 10x + 8 = (3x + 4)(x + 2) · 6x^2 - 5x - 6 = (3x + 2)(2x - 3)
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.
Explanation
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.
In simple words
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.
Key points
- product q
- sum p
- ax^2 + bx + c
- ac
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Splitting the Middle Term
Sticky Notes · Quick Revision
WHAT
- 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.
Key Points
- 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.
HOW
- 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.
Examples
- x2 + 5x + 6 = (x + 2)(x + 3)
- x2 - 5x + 6 = (x - 2)(x - 3)
- x2 + x - 6 = (x + 3)(x - 2)
- 2x2 + 7x + 3 = (2x + 1)(x + 3)
- 3x2 + 10x + 8 = (3x + 4)(x + 2)
Common Mistake
- For a !
- = 1, the product you split is ac, not c.
Exam Tip
- 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.
60-second revision
- 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.
- For a !
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