No need to solve the equation: The sum and product of a quadratic's roots are hidden in its coefficients:
for ax² + bx + c = 0, sum = −b/a and product = c/a. You never have to find the roots.
IB-style question — sum and product
The equation 2x² − 6x + 1 = 0 has roots α and β.
Find α + β and αβ.
Step by step
- Read off a = 2, b = −6, c = 1. Sum = −b/a.
- Product = c/a.
Final answer
α + β = 3 and αβ = ½.
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Same idea, all the way up: For any polynomial aₙxⁿ + … + a₀ = 0:
sum of roots = −aₙ₋₁/aₙ and product = (−1)ⁿ a₀/aₙ.
For a cubic ax³ + bx² + cx + d: sum = −b/a, product = −d/a.
IB-style question — a cubic
The roots of x³ − 4x² + x + 6 = 0 are α, β, γ.
Find α + β + γ and αβγ.
Step by step
- Here a = 1, b = −4, d = 6. Sum = −b/a.
- Product = (−1)³ d/a = −d/a.
Final answer
Sum = 4, product = −6.