a-maths

Since α , β are the roots of x² - 3x + p = 0

So, product of roots, αβ = p

Sum of roots, α + β = 3

α^4 + β^4 = 17

(α² + β²)² - 2α²β² = 17

[(α + β)² - 2αβ]² - 2(αβ)² = 17

[(3)² - 2p]² - 2p² = 17

81 - 36p + 4p² - 2p² = 17

p² - 18p + 32 = 0

(p - 16)(p - 2) = 0

p = 2 or 16

consider α+β=3
αβ=p

(α+β)^4=α^4 +4α^3β +6α^2β^2 +4αβ^3 +β^4
α^4+β^4=(α+β)^4 -4α^3β -6(αβ)^2 -4αβ^3
=(α+β)^4 -6(αβ)^2 -4αβ(α^2+β^2)
=(α+β)^4 -6(αβ)^2 -4αβ[(α+β)^2-2αβ]
=81-6p^2 -4p(9-2p)
=2p^2-36p+81

α^4+β^4 =17
2p^2-36p+81=17
2p^2-36p+64=0
p=16 or2