Question
Form a quadratic equation whose roots are 3 and −2.
Solution — Step by Step
If α and β are roots of a quadratic, then the equation is (x − α)(x − β) = 0. We’re told α = 3 and β = −2, so we write (x − 3)(x − (−2)) = 0, which simplifies to (x − 3)(x + 2) = 0.
Multiply the two factors using FOIL:
Combine like terms: x² − x − 6.
Setting this equal to zero gives us:
This is the required quadratic equation.
Substitute x = 3: 9 − 3 − 6 = 0 ✓
Substitute x = −2: 4 + 2 − 6 = 0 ✓
Both roots satisfy the equation — we’re done.
Why This Works
Any quadratic with roots α and β can be written as k(x − α)(x − β) = 0 where k is any non-zero constant. We take k = 1 for the simplest form. This is called the factor form of a quadratic.
There’s a faster path using Vieta’s formulas. The sum of roots is α + β = 3 + (−2) = 1 and the product is αβ = 3 × (−2) = −6. The standard form is:
Plugging in: x² − (1)x + (−6) = 0, which gives x² − x − 6 = 0. Same answer, faster route.
This Vieta’s approach is the one to master — it’s significantly faster in JEE Main MCQs where you’re given sum/product directly without naming the individual roots.
Alternative Method — Vieta’s Formulas Directly
Sum of roots = 3 + (−2) = 1
Product of roots = 3 × (−2) = −6
Substituting:
Two lines, same answer. In CBSE board exams, both methods earn full marks — but Vieta’s is cleaner when the roots are messy (like 2 + √3 and 2 − √3).
Common Mistake
Sign error on the second root. The factor for root β = −2 is (x − β) = (x − (−2)) = (x + 2), NOT (x − 2). Writing (x − 3)(x − 2) = 0 gives the wrong equation x² − 5x + 6 = 0 with roots 3 and 2. Always subtract the root value, and be careful with negatives.
This exact error cost students marks in CBSE 2024 — the question appeared with roots 3 and −2 specifically to test this sign awareness.
After writing your equation, spend 10 seconds verifying by substituting both roots back. In a 1-mark board question, that verification catches the sign error before the examiner does.