Maths problem how to do, thanks?

兩題都不能完全算出結果, 只能用 k 的函數表示.


Let k be a constant. If the roots of the
quadratic equation x^2-5x+k = 0 are α and β,
find α^3β+5αβ^2.

[Sol.]
α^3β+5αβ^2
    = αβ(α^2-5α+k)+5α^2β-kαβ+5αβ^2
    = 5αβ(α+β)-kαβ
    = 5k(5) - k(k)
    = 25k-k^2 = k(25-k)


If α≠β and {6α-α^2 = k, 6β-β^2 = k},
find (α-β)^2.

[Sol.]
α, β are solutions of equation
    x^2 - 6x + k = 0
So,
    α + β = 6, αβ = k

 (α-β)^2 = (α+β)^2-4αβ
      = 6^2 - 4k
      = 36 - 4k

α ≠ β if and only if k ≠ 9.