### Theory:

We have learnt that the roots of the quadratic equation $$ax^2 + bx + c = 0$$ can be found by the quadratic formula:

$x=\frac{-b±\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$

Important!
$$b^2 - 4ac$$ is called the discriminant of the quadratic equation $$ax^2 + bx + c = 0$$. It is denoted by the letter $$\Delta$$ or $$D$$.
Let us discuss the nature of the roots of the quadratic equation depending on the discriminant.

Case I: $$\Delta = b^2 - 4ac > 0$$

Here, $$b^2 - 4ac > 0$$. That means the value of the discriminant is positive.

Then, the possible roots are $\frac{-b+\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$ and $\frac{-b-\sqrt{{b}^{2}-4\mathit{ac}}}{2a}$.
If $$\Delta = b^2 - 4ac > 0$$, then the roots are real and distinct.

Case II: $$\Delta = b^2 - 4ac = 0$$

Here, $$b^2 - 4ac = 0$$. That means the value of the discriminant is zero.

$x=\frac{-b+\sqrt{0}}{2a}$ and $x=\frac{-b-\sqrt{0}}{2a}$

$x=\frac{-b}{2a}$ and $x=\frac{-b}{2a}$

The possible roots are $\frac{-b}{2a}$ and $\frac{-b}{2a}$.
If $$\Delta = b^2 - 4ac = 0$$, then the roots are real and equal.

Case III: $$\Delta = b^2 - 4ac < 0$$

Here, $$b^2 - 4ac < 0$$. That means the value of the discriminant is negative.

We won't get any real roots in this case.
If $$\Delta = b^2 - 4ac < 0$$, then there are no real roots.
Relation between roots and coefficients of a quadratic equation
If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$ax^2 + bx + c = 0$$, then:

(i) Sum of the roots $$=$$ $\mathrm{\alpha }+\mathrm{\beta }=\frac{-b}{a}$

(ii) Product of the roots $$=$$ $\mathrm{\alpha }\mathrm{\beta }=\frac{c}{a}$
Quadratic equation $$=$$ $$x^2 - (\text{sum of the roots})x + \text{product of the roots}$$
Important!
To know more about the relationship between roots and coefficients of a equation click here.