Question

Solve the equation 6x

4 − 3x

3 + 8x

2 − x + 2 = 0 being given that it has a

pair of roots whose sum is zero.​

Answer 1

Answer:

The given equation is 6x4−3x3+8x2−x+2=0

Given: α+β=0 Hence, α=−β

Now, from the given equation

α+β+γ+δ=3/6=1/2

(α+β)(γ+δ)+αβ+γδ=8/6=4/3

(α+β)γδ+αβ(γ+δ)=1/6

αβγδ=2/6=1/3

Applying the given condition, the above equations become,

γ+δ=1/2 ---(I)

αβ+γδ=4/3 ---(II)

αβ(γ+δ)=1/6 ---(III)

αβγδ=1/3 ---(IV)

using I and III, we get, αβ=1/3 ---(V)

using IV and V, we get, γδ=1 ---(VI)

So, now we have 2 quadratic equations,

x2−0.x+1/3=0 (for α,β) and x2−1/2.x+1=0 (for γ,δ)

ie: 3x2+1=0 and 2x2−x+2=0

Solving, the quadratic equations, we get, the roots as

i3–√,−i3–√,1+i1–√54,1−i1–√54