Question

Let \(a, b, c\) be real numbers such that \(a + b + c < 0\) and the quadratic equation \(ax^2 + bx + c = 0\) has imaginary roots. Then

Answer 1

\(ax^2 + bx + c = 0\) has imaginary roots
\(\Rightarrow\) \(f(x)=ax^2 + bx + c \) does not intersect \(x\)-axis for any real \(x\). Thus, its graph is either an upward opening parabola lying completely above the \(x\)-axis or its a downward opening parabola lying completely below the \(x\)-axis.

Since, \(f(1)=a+b+c<0\).
So, it lies completely below the \(x\)-axis.
\(\Rightarrow a<0\)
Also \(D<0\)
\(\Rightarrow b^2-4ac<0\)
\(\Rightarrow b^2<4ac\)
\(\Rightarrow a \) and \(c\) will have the same sign.