What will be the nature of the roots of the equation?

Question

What will be the nature of the roots of the equation $7x^2 – 2x + 3 = 0$

If I have two options like Imaginary and complex. Are both options are valid? Or more near answer is Imaginary. In my book it is written as if Discriminate is less than $0$ then roots are Imaginary? So should I go with the option Imaginary?

Answer 1

Every quadratic equation is of the form $ax^2+bx+c=0.$

In this case, the discriminant, $\Delta=b^2-4ac=(-2)^2-4(7)(3)-80<0,$ so the roots are complex. (By the way, all imaginary numbers are complex.)

Here are the 3 cases in general (for a quadratic equation):

• If $\Delta>0,$ then the roots are real and distinct (e.g. $4$ and $-2$)
• If $\Delta=0$, then the roots are real and repeated (e.g. $1$ and $1$)
• If $\Delta<0$, then the roots are complex (e.g. $3 \pm i, \pm i,$ etc.), and, if the coefficients of the equation are real, then the roots of the equation appear in conjugate pairs (i.e. the roots are of the form $\alpha\pm\beta i$ where $\alpha $ and $\beta $ are real).

Incidentally, just to show, visually, which numbers belong to which set, here's a useful picture I stole from the internet: