Question

Which equation has imaginary roots?

Answer 1

Here we will assume a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ and take three cases relating to the discriminant value given as $D={{b}^{2}}-4ac$. In the first case we will consider D > 0 then in the second case D = 0 and in the third case D < 0. We will check the cases where the roots are imaginary or real.

Complete step-by-step solution:
In mathematics imaginary roots are of the form $ip$ where p is a real number and $i$ is the imaginary number $\sqrt{-1}$ which is the solution of the quadratic equation ${{x}^{2}}+1=0$. The imaginary roots appear in the quadratic equation. We know that a quadratic equation is of the form $a{{x}^{2}}+bx+c=0$, it has two roots and its discriminant value is given as $D={{b}^{2}}-4ac$. There are three possible cases that may arise, let us check them one by one.
(1) If D > 0 then the two roots of the quadratic equation are real and distinct.
(2) If D = 0 then the two roots of the quadratic equation are real and equal.
(3) If D < 0 then the two roots of the quadratic equation are imaginary or complex.

Note: Note that if the root is of the form $q+ip$, where q and p are real numbers, then it is called a complex root instead of simply an imaginary root. Both imaginary and real numbers are a subset of complex numbers. In case of two real and distinct roots the graph of the quadratic equation which is a parabola cuts the x – axis at two different points. In case of two real and equal roots the x – axis acts as a tangent to the parabola and in case of two complex roots the parabola either lies completely above the x – axis or below the x – axis.