If α and β are the roots of the quadratic equation x2+(p-3) x-2 p=3(p ∈ R), then the minimum value of (α2+β2+α β), is

Question

If α and β are the roots of the quadratic equation x2+(p-3) x-2 p=3(p ∈ R), then the minimum value of (α2+β2+α β), is (A) 2 (B) 4 (C) 8 (D) 12

Tardigrade Admin · Accepted Answer

α+β=-(p-3) α β=-2 p-3 ∵ α2+β2+α β= (α+β)2-α β =(p-3)2+2 p +3 = p 2-4 p +12 =( p -2)2+8 ∴ minimum value of α2+β2+α β will be 8 .

Q. If $α$ and $β$ are the roots of the quadratic equation $x^{2} + (p - 3) x - 2 p = 3 (p \in R)$ , then the minimum value of $(α^{2} + β^{2} + α β)$ , is