Numerical answer
α and β are the roots of x² − 7x + k = 0. If α² + β² = 29, find the value of α³ + β³.
Show answer and solution
Answer: 133
α + β = 7 and αβ = k.
α² + β² = (α + β)² − 2αβ = 49 − 2k = 29, so k = 10 (the roots are 2 and 5).
α³ + β³ = (α + β)³ − 3αβ(α + β) = 343 − 3 × 10 × 7 = 343 − 210 = 133.
Check: 2³ + 5³ = 8 + 125 = 133. ✓
α² + β² = (α + β)² − 2αβ = 49 − 2k = 29, so k = 10 (the roots are 2 and 5).
α³ + β³ = (α + β)³ − 3αβ(α + β) = 343 − 3 × 10 × 7 = 343 − 210 = 133.
Check: 2³ + 5³ = 8 + 125 = 133. ✓
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