Let f(x)=x2+b1x+c1,g(x)=x2+b2x+c2. Let the real roots of f(x)=0 be α,β and real roots of g(x)=0 be α+h,β+h.The least value of f(x) is −1/4. The least value of g(x) occurs at x = 7/12.The least value of g(x) is The value of b2 isThe roots of f(x) = 0 are

(β−α)=((β+h)−(α+h))(β+α)2−4αβ=[(β+h)+(α+h)]2−4(β+h)(α+h)−b12−4c1=−b22−4c2D1=D2The least value of f(x) is−D14=−14⇒D1=1 and D2=1Therefore, the least value ofg(x) is −D24=−14The least value of g(x) occurs at −b22=72 or b2=−7⇒ b22−4c2=D2 or   49−4c2=1 or 484=c2 or c2=12 ⇒ x2−7x+12=0 or x=3,4(β−α)=((β+h)−(α+h))(β+α)2−4αβ=[(β+h)+(α+h)]2−4(β+h)(α+h)−b12−4c1=−b22−4c2D1=D2The least value of f(x) is−D14=−14⇒D1=1 and D2=1Therefore  the least value of g(x) is −D24=−14The least value of g(x) occurs at −b22=72 or b2=−7 ⇒ b22−4c2=D2 or   49−4c2=1 or 484=c2 or c2=12 ⇒ x2−7x+12=0 or x=3,4   (β−α)=((β+h)−(α+h))(β+α)2−4αβ=[(β+h)+(α+h)]2−4(β+h)(α+h)−b12−4c1=−b22−4c2D1=D2The least value of f(x) is−D14=−14⇒D1=1 and D2=1Therefore least value ofg(x) is −D24=−14The least value of g(x) occurs at −b22=72 or b2=−7 ⇒ b22−4c2=D2 or   49−4c2=1 or 484=c2 or c2=12 ⇒ x2−7x+12=0 or x=3,4