If f(x)=ax2+bx+c,  g(x)=−ax2+bx+c,  where ac≠0,  then the minimum number of real roots of the equation f(x)  g(x)=0  is___.

Let D1  and D2 be the discriminates of −ax2+bx+c=0  respectively. Then D1=b2−4ac  and D2=b2+4ac. Now ac≠0⇒ either ac>0  or ac<0. If ac>0, then D2>0.  Therefore, the roots of −ax2+bx+c=0  are real.If ac<0,  then D1>0.  Therefore, the roots of ax2+bx+c=0  are real.Thus, f(x)g(x)  has atleast two real roots.