If α, β be the zeroes of the polynomial 2x2 + 5x + k such that (α + β)2 – αβ = 21/4 , then k = ? ​

Question

If α, β be the zeroes of the polynomial 2x² + 5x + k such that (α + β)² – αβ = \(\frac{21}4\) , then k = ? 

(a) 3 

(b) -3 

(c) -2 

(d) 2

Answer 1

(d) 2 

Since α and β are the zeroes of 2x² + 5x + k, 

we have:

α + β = \(\frac{-5}2\) and αβ = \(\frac{k}2\)

Also, it is given that α² + β² + αβ = \(\frac{21}4\)

⇒ (α + β)² – αβ = \(\frac{21}4\)

⇒ \((\frac{-5}2)^2\) - \(\frac{k}2\) = \(\frac{21}4\)

⇒ \(\frac{25}4\)  - \(\frac{k}2\) = \(\frac{21}4\)

⇒ \(\frac{k}2\) = \(\frac{25}4\) - \(\frac{21}4\) = \(\frac{4}4\) = 1

⇒ k = 2