Concept 4.2
Factorisation: Split the Middle Term
Source: NCERT Mathematics, Class 1012 minute lesson
Formal statement
To solve ax² + bx + c = 0 by factorisation, split the middle term bx into two terms whose coefficients multiply to a × c and add to b, factorise, and set each factor equal to zero.
The intuition
For 2x² − 5x + 3 = 0 you need two numbers that multiply to a × c = 6 and add to −5: −2 and −3. Then 2x² − 2x − 3x + 3 = (2x − 3)(x − 1).
Each factor set to zero gives one root: x = 3/2 or x = 1. A product is zero only when a factor is zero, which is what lets one hard equation become two easy ones.
The attention
Look at the product a × c, not at c alone.
2x² − 5x + 3 → need product a × c = 6, sum −5
Look hereNot here
- Look here
- The product a × c. The two numbers you need multiply to a × c. For 2x² − 5x + 3 that is 6, giving −2 and −3.
- Not here
- The constant c alone. Using c = 3 alone only works when a = 1. Here no pair multiplies to 3 and adds to −5, and you get stuck.
- What it teaches
- When a shortcut stops working, check which special case it was built for.