Concept 2.3
Sum and Product of Zeroes
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then α + β = −b/a and αβ = c/a.
The intuition
Take x² − 5x + 6 = (x − 2)(x − 3). Its zeroes are 2 and 3; their sum is 5 and their product is 6. The 5 shows up in the polynomial with a minus sign, the 6 without.
That minus is not an accident. Expanding (x − α)(x − β) gives x² − (α + β)x + αβ: the sum always arrives with a minus attached.
The attention
Look at the minus sign in the sum of zeroes.
α + β = −b/a αβ = c/a
Look hereNot here
- Look here
- The minus sign in the sum. The sum of zeroes is −b/a. Dropping that minus is the most common slip, and it flips the answer.
- Not here
- The product, c/a. The product c/a has no minus. Copying its pattern onto the sum is exactly how the sign gets lost.
- What it teaches
- When two formulas look alike, look at where they differ.