If the product of two of the four roots of the biquadratic equation X⁴+18x³+kx²+200x+1984 =0 is -32. Find a and b.

Question

If the product of two of the four roots of the biquadratic
equation  X4+18x3+kx2+200x+1984 =0 is -32
Find a and b

Koka Sri Lakshmi Divya Sai · Accepted Answer

the correct question is

The sum of two roots of the biquadratic equation is $x^{4} - 18 x^{3} + k x^{2} + 200 x - 1984 = 0$ -32,find k

$Given x^{4} - 18 x^{3} + k x^{2} + 200 x - 1984 = 0 L e t α, β, γ, δ be the roots of the given equation α + β + γ + δ = 18 - - - - (i) α β + β γ + γ δ + α γ + α δ + β δ = k - - - (i i) α β γ + β γ δ + α γ δ + α β δ = - 200 - - - (i i i) α β γ δ = - 1984 - - (i v) Given α β = - 32 From (iv) we get \Rightarrow γ δ = 62 From (iii) we get - 32 γ + 62 β + 62 α - 32 δ = - 200 \Rightarrow 62 α + 62 β - 32 γ - 32 δ = - 200 \Rightarrow 31 α + 31 β - 16 γ - 16 δ = - 100 - - - (v) Solving (i) and (v) we get α + β = 4 - - - (v i) From (i) we get γ + δ = 14 (α - β)^{2} = (α + β)^{2} - 4 α β \Rightarrow (α - β)^{2} = (4)^{2} - 4 (- 32) \Rightarrow (α - β)^{2} = 16 + 128 \Rightarrow (α - β)^{2} = 144 \Rightarrow α - β = 12 - - (v i i) Solving (vi) and (vii) we get α = 8 and β = - 4 By substituting these values in (ii) we get α β + β γ + γ δ + α γ + α δ + β δ = k \Rightarrow - 32 + β (γ + δ) + α (γ + δ) + 62 = k \Rightarrow - 32 + (- 4) (14) + 8 (14) + 62 = k \Rightarrow - 32 - 56 + 112 + 62 = k \Rightarrow k = 86$

If the product of two of the four roots of the biquadratic equation X4+18x3+kx2+200x+1984 =0 is -32. Find a and b.

If the product of two of the four roots of the biquadratic equation X⁴+18x³+kx²+200x+1984 =0 is -32. Find a and b.