√3+ √2
If a =
√3- 82
V3-2
& b =
V3+2
, find a2 + b 2
Answers
Step-by-step explanation:
Question :
If α and β are the zeros of the polynomial p(x) = 2x² + 5x + k satisfying satisfying the relation α²+ β²+αβ = ²¹/₄ , then find the value of k
Solution :
On comparing given polynomial 2x²+5x+k with ax²+bx+c , we get ,
➙ a = 2 , b = 5 , c = k
Sum of zeroes , α + β = - ᵇ/ₐ
⇒ α + β = - ⁵/₂ \pink{\bigstar}★
Product of zeroes , αβ = ᶜ/ₐ
⇒ αβ = ᵏ/₂ \green{\bigstar}★
Given that ,
\rm \alpha ^2 + \beta ^2 + \alpha \beta = \dfrac{21}{4}α
2
+β
2
+αβ=
4
21
\begin{gathered}\bullet\ \; \sf \red{(x+y)^2=x^2+y^2+2xy}\\\\ \bullet\ \; \sf \red{x^2+y^2= (x+y^2)-2xy}\end{gathered}
∙ (x+y)
2
=x
2
+y
2
+2xy
∙ x
2
+y
2
=(x+y
2
)−2xy
:\implies \rm ( \alpha + \beta )^2-2 \alpha \beta + \alpha \beta = \dfrac{21}{4}:⟹(α+β)
2
−2αβ+αβ=
4
21
:\implies \rm ( \alpha + \beta )^2- \alpha \beta = \dfrac{21}{4}:⟹(α+β)
2
−αβ=
4
21
Sub. values ,
:\implies \rm \bigg( - \dfrac{5}{2} \bigg)^2 - \dfrac{k}{2} = \dfrac{21}{4}:⟹(−
2
5
)
2
−
2
k
=
4
21
:\implies \rm \dfrac{25}{4}- \dfrac{k}{2} = \dfrac{21}{4}:⟹
4
25
−
2
k
=
4
21
:\implies \rm \dfrac{25-2k}{4} = \dfrac{21}{4}:⟹
4
25−2k
=
4
21
:\implies \rm 25-2k=21:⟹25−2k=21
:\implies \rm 25-21 = 2k:⟹25−21=2k
:\implies \rm 4=2k:⟹4=2k
:\implies \rm 2=k:⟹2=k
:\implies \rm k=2\ \; \blue{\bigstar}:⟹k=2 ★
Value of k is 2 .