If α, β are the roots of x2+7 x+9=0, then the value of [((α/3+β))2+((β/3+α))2] is [Note: [k] denotes greatest integer function less than or equal to k.]

Question

If α, β are the roots of x2+7 x+9=0, then the value of [((α/3+β))2+((β/3+α))2] is [Note: [k] denotes greatest integer function less than or equal to k.] (A) 16 (B) 15 (C) 18 (D) 17

Tardigrade Admin · Accepted Answer

x2+6 x+9=-x (x+3)2=-x ⇒ (α2/(β+3)2)+(β2/(α+3)2)=-((α2/β)+(β2/α)) =-(((α+β)3-3 α β(α+β)/α β))=(154/9) ⇒[((α/3+β))2+((β/3+α))2]=17

Q. If α,β are the roots of x2+7x+9=0, then the value of [(3+βα​)2+(3+αβ​)2] is [Note: [k] denotes greatest integer function less than or equal to k.]

16

15

18

17

x2+6x+9=−x (x+3)2=−x ⇒(β+3)2α2​+(α+3)2β2​=−(βα2​+αβ2​) =−(αβ(α+β)3−3αβ(α+β)​)=9154​ ⇒[(3+βα​)2+(3+αβ​)2]=17

Q. If $α, β$ are the roots of $x^{2} + 7 x + 9 = 0$ , then the value of $[(\frac{α}{3 + β})^{2} + (\frac{β}{3 + α})^{2}]$ is
[Note: [k] denotes greatest integer function less than or equal to k.]