Question

the sum of two roots of the equation x² - ax + b = 0 equal to k times the different of the roots. Show that a²(k²-1)= 4k²b

Answer 1

Let α and β is the roots of x² - ax + b = 0
A/c to question,
sum of roots = k ( difference of roots )
(α + β) = k(α- β)
take square both sides,
(α + β)² = k²(α -β)²

But we know,
Sum of roots = α + β = -{coefficient of x/coefficient of x²} = a

product of roots = αβ = {constant/coefficient of x²} = b

(α + β)² = k² (α - β)²
(a)²= k² (α² + β² -2αβ)
a² = k²{(α + β)² -4αβ}
a² = k²{a² - 4b}
a² - a²k² = -4bk²
a²k² - a² = 4k²b
a²(k²-1) = 4k²b
Hence, proved