Sums & Products of Roots
Take the quadratic ax² + bx + c = 0. Divide every term by a (assuming a ≠ 0) to get the monic form:
If α and β are the roots, the quadratic can also be written in factorised form:
Both versions describe the same polynomial, so their coefficients must match. Equating gives the two golden rules:
Sum of Roots
α + β = −b/a
Note the negative sign — easy to forget!
Product of Roots
αβ = c/a
No sign change here — just c over a.
Classic trap: Students forget the negative on the sum. If your quadratic is 5x² − 30x − 3 = 0, the sum of roots is −(−30)/5 = +6, not −6. Always ask: "is b positive or negative first?"
Memory trick: Think of a monic quadratic as x² − (SUM)·x + (PRODUCT) = 0. The sum appears with a minus sign; the product appears as-is.
Not in the formula booklet — you must memorise both α + β = −b/a and αβ = c/a.
Worked Example
Example 1 — Reading off sum and product
The equation 5x² − 30x − 3 = 0 has roots α and β. Find α + β and αβ without solving.
- Identify a = 5, b = −30, c = −3.
- Sum: α + β = −b/a = −(−30)/5 = 6
- Product: αβ = c/a = (−3)/5 = −3/5
Practice Question
The equation 4x² + 7x − 2 = 0 has roots α and β. Find the values of α + β and αβ without solving the equation.
Expressions Involving the Roots
Once you know α + β and αβ, you can evaluate any symmetric expression in α and β. The strategy is always the same: rewrite the expression using identities, then substitute in your known values.
The Identities You Must Know
None of these are in the formula booklet — but you only need to remember how they're derived, not memorise them blindly. Understanding the derivation means you can rebuild them under pressure.
Quick Derivation
→ Expand: (α + β)² = α² + 2αβ + β²
→ Rearrange: α² + β² = (α + β)² − 2αβ ✓
Quick Derivation
→ Expand: (α + β)³ = α³ + 3α²β + 3αβ² + β³
→ Factor middle terms: = α³ + β³ + 3αβ(α + β)
→ Rearrange: α³ + β³ = (α + β)³ − 3αβ(α + β) ✓