Question
Find the zeroes of the polynomial p(x) = x² - 5x + 6 and verify the relationship between the zeroes and the coefficients.
Solution — Step by Step
We need two numbers that multiply to give 6 (the constant term) and add to give −5 (the coefficient of x). Think: what pairs multiply to 6? We have (1, 6), (2, 3), (−1, −6), (−2, −3).
The pair (−2, −3) works: (−2) × (−3) = 6 and (−2) + (−3) = −5. This is the split we need. We rewrite the middle term using these numbers.
Group the terms:
Set each factor to zero. p(x) = 0 gives us (x − 2)(x − 3) = 0, so:
The zeroes are α = 2 and β = 3.
For p(x) = x² − 5x + 6, we have a = 1, b = −5, c = 6.
Both relations check out.
Why This Works
A quadratic polynomial ax² + bx + c can always be factorised into a(x − α)(x − β) where α and β are its zeroes. When we expand this, the sum of zeroes shows up as −b/a and the product as c/a — this is a consequence of Vieta’s formulas.
The split-the-middle-term method works because we’re essentially reversing the FOIL expansion. We search for two numbers that reconstruct the original middle term once split and grouped.
This verification step isn’t optional in NCERT — it’s worth full marks in board exams. Skipping it loses you 1–2 marks even if your zeroes are correct.
Alternative Method
We can use the quadratic formula directly:
This gives x = 3 or x = 2 — same result. For easy quadratics like this one, factorisation is faster. Save the formula for when the discriminant isn’t a perfect square.
If b² − 4ac is a perfect square, the polynomial factors nicely over integers. If not, the zeroes are irrational — and you’ll need the formula or completing the square. Always check the discriminant first to pick your method.
Common Mistake
Students often write the zeroes as x = −2 and x = −3 after finding the pair (−2, −3). This is the classic sign-flip error. The factors are (x − 2) and (x − 3), not (x + 2) and (x + 3). Setting x − 2 = 0 gives x = +2, not −2. The numbers you split the middle term with are not the zeroes directly.