If the equation x2+2 α x+α2-1=0 and x2+2 β x+β2-1=0 have a common root (α ≠ β) then the value of the expression 2 α2-4 α β-|α-β|+|α-β| β2, is

Question

If the equation x2+2 α x+α2-1=0 and x2+2 β x+β2-1=0 have a common root (α ≠ β) then the value of the expression 2 α2-4 α β-|α-β|+|α-β| β2, is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Subtracting the two equation we get the common root as x=-(1/2)(α+β). Substituting this in any equation we get (1/4)(α+β)2-(2 α(α+β)/2)+α2-1=0 ⇒ (1/4)(α+β)2-α β-1=0 ⇒ (1/4)[(α+β)2-4 α β]-1=0 ⇒ (1/4)[(α-β)2]=1 ⇒ |α-β|=2 text now 2 α2-4 α β-|α-β|+|α-β| β2 =2 α2-4 α β+2 β2-2 =2(α-β)2-2=2 ⋅ 4-2=8-2=6

Q. If the equation x2+2αx+α2−1=0 and x2+2βx+β2−1=0 have a common root (α=β) then the value of the expression 2α2−4αβ−∣α−β∣+∣α−β∣β2, is

Q. If the equation $x^{2} + 2 αx + α^{2} - 1 = 0$ and $x^{2} + 2 β x + β^{2} - 1 = 0$ have a common root $(α \neq = β)$ then the value of the expression $2 α^{2} - 4 α β - ∣ α - β ∣ + ∣ α - β ∣ β^{2}$ , is