Single correct option
If α and β are the roots of x² − 6x + k = 0 and α² + β² = 20, what is the value of α³ + β³?
- A. 64
- B. 72
- C. 80
- D. 96
Show answer and solution
Answer: (B) 72
α + β = 6 and αβ = k.
α² + β² = (α + β)² − 2αβ = 36 − 2k = 20 → k = 8.
α³ + β³ = (α + β)³ − 3αβ(α + β) = 216 − 3 × 8 × 6 = 216 − 144 = 72.
Check: the roots are 2 and 4; 8 + 64 = 72 ✓.
Tip: Symmetric functions of roots are always expressible through the sum and product; you rarely need the roots themselves.
α² + β² = (α + β)² − 2αβ = 36 − 2k = 20 → k = 8.
α³ + β³ = (α + β)³ − 3αβ(α + β) = 216 − 3 × 8 × 6 = 216 − 144 = 72.
Check: the roots are 2 and 4; 8 + 64 = 72 ✓.
Tip: Symmetric functions of roots are always expressible through the sum and product; you rarely need the roots themselves.
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