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

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

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.