If, we have any quadratic equation of the form , where a, b, c are real numbers and then how can we determine about the nature of roots of such quadratic equation?

**Answer:** We can determine about nature of roots of any quadratic equation through discriminant. Now, the question arises what is discriminant.

We often use quadratic formula, to find roots of any given quadratic equation. Sub-part () of quadratic formula is called discriminant of quadratic equation.

Therefore, discriminant of any quadratic equation =

Lets take an example, we have quadratic equation, , if we compare it with general form of quadratic equation , we get

and .

Discriminant = =

Similarly, we can find discriminant of other quadratic equations.

**Now, the question is how can we determine nature of roots from the value of discriminant of quadratic equation.**

**If, Discriminant >0, then two roots of quadratic equation are distinct and real.**

**If, Discriminant =0, then two roots of quadratic equation are real and equal.**

**If, Discriminant < 0, then there are no real roots for given quadratic equation.**

Lets take three different examples, one for each case. Suppose, we have three quadratic equations:

**(1) **

**(2) **

**(3) **

We will now determine nature of roots of these three quadratic equations using discriminant.

**(1) **

Comparing this equation with general form , we get and .

Discriminant =

Therefore, discriminant of equation is greater than 0. Therefore, equation has real and distinct roots. You can also verify this by actually finding roots of equation. Roots of equation will come out be and which are distinct and real numbers.

**(2) **

Comparing this equation with general form , we get and .

Discriminant =

Therefore, discriminant of equation is equal to zero. Therefore, equation has equal and real roots. You can also verify this by actually finding the roots of equation. Roots will come out to be and which are equal and real numbers.

**(3) **

Comparing this equation with general form , we get and .

Discriminant =

Therefore, discriminant of equation is less than zero. Therefore, equation has no real roots. You can also verify this by actually trying to find the roots of equation. When, you will apply quadratic formula, to find roots of equation, you will get in square root. We all know that square of negative number does not exist. Therefore, there will be no solution for this quadratic equation. Or, you can say there is no real solution for this quadratic equation.

**Note*:** Solution to such quadratic equations can be complex but it is beyond the scope of books that we study in tenth grade. So, you do not need to worry about this. You just need to write no solution or no real solution for such quadratic equations.