The roots of quadratic equation y2 – 8y + 14 = 0 are α and β. Find the value of (1 + α + β2)( 1 + β + α2)
Explanation:
The roots of the equation y2 – 8y + 14 = 0 are α and β.
∴ Sum of roots = α + β = −(−8)/1 = 8 and Product of roots = αβ = (14)/1 = 14
∴ (1 + α + β2) (1 + β + α2) = 1 + α + β2 + β + αβ + β3 + α2 + α3 + α2β2
= 1 + (α + β) + (αβ) + (αβ)2 + (α2 + β2) + (α3 + β3)
= 1 + (α + β) + (αβ) + (αβ)2 + (α + β)2 – (2αβ) + (α + β)3 – (3αβ)(α + β)
= 1 + 8 + 14 + (14)2 + (8)2 – 2(14) + (8)3 – 3(14)(8)
= 23 + 196 + 64 – 28 + 512 – 336 = 431
Hence, option (b).
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