Question

if alpha and beta are zeroes of the polynomial p(x)=2x^2+5x+k=0.satisfying the relation square of alpha +square of beta +alpha×beta=21/4. then find the value of k

Answer 1

A and B are the zeroes
therefore A + B = -5/2
therefore (A + B)^2 = (-5/2)^2  = 25/4.............. (i)

also A * B = k /2.............(ii)
 now,(A+ B)^2 - A * B = A^2 + B^2 + A * B
therefore, 25 /4- k/2 = 21/4
25 - 2k = 21
2k = 4
k = 2

Answer 2

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Since, α and β are the zeroes of the polynomial p(x)=2x²+5x+k=0.

α + β = -b/a ⇒ -5/2

and  

αβ = c/a ⇒ k/2

α² + β² + αβ = 21/4  ( given )

⇒ (α² + β² + 2αβ ) - αβ = 21/4

⇒ (α + β )² - αβ = 21/4

⇒ 25/4 - k/2 = 21/4   [ α+β = -5/2 and αβ =k/2 ]

⇒ -k/2 = -1

⇒ k = 2.

Hence, the value of "k" = 2.