Question

if one root is equal to the square of the other root of the equation x2+x-k=0,what is the value of k?​

Answer 1

Answer:

Step-by-step explanation:

Given :

If one root is equal to the square of the other root of the equation

x² + x - k = 0 ,

what is the value of k?​

Solution :

Let the roots be α & β,.

We know that,

for an equation of the form,

ax² + bx + c = 0,.

⇒ Sum of the roots = $\frac{-b}{a}$

⇒ α + β = $\frac{-b}{a}$

⇒ Product of the roots = $\frac{c}{a}$

⇒ αβ = $\frac{c}{a}$

&

One root is square of the other root,.

⇒ β = α²

Hence,

α + α² = $\frac{-(1)}{1}$ = -1

⇒ α² + α = -1

⇒ α² + α = -1

⇒ α² + α +1 = 0..(i)

⇒ α (α)² = -k

⇒ α³ = -k ...(ii)

By solving (i),

by solving for value of α in,

α² + α + 1 = 0

a = 1 , b = 1 , c = 1

AS, the discriminant is < 0,.

b² - 4ac = 1² - 4 × 1 × 1 = 1 -- 4 =  -3 < 0,.

There is no real value exist for α,

hence k doesn't posses any real value since, k = -α³