The nature of roots of a quadratic equation can be determined by the value of its discriminant (D). See the table below.

First calculate the discriminant by the following formula:

If D equals zero, there are two real and equal roots. In other words, there is only one solution to the quadratic equation. In such a case, the graph of the quadratic equation touches the x-axis at one point.

If D is a negative real number (lesser than zero), then the roots are

`D = b^2 - 4ac`

`D = 6^2 - 4(1)(5)`

`D = 16`

Since D is greater than zero thus there are two real and distinct roots. Further, 16 is a

Here is a graph of the above equation. It has two x-intercepts, -1 and -5, which are its roots or solutions. Notice that -1 and -5 are a pair of real, distinct and rational numbers. This coincides with our observation above the nature of roots above.

`D = b^2 - 4ac`If D is a positive real number (greater than zero), then there are two real and distinct roots of the quadratic equation. "real" means that the roots are

__real numbers__and "distinct" means that they are not equal (that is, there are two different roots, not two same roots). In this case, the graph of the equation intersects the x-axis at two distinct points.If D equals zero, there are two real and equal roots. In other words, there is only one solution to the quadratic equation. In such a case, the graph of the quadratic equation touches the x-axis at one point.

If D is a negative real number (lesser than zero), then the roots are

__imaginary or complex numbers__. This means that they are of the form `ai + b`, where 'i' is the imaginary number. This is also quite logical, since in the quadratic formula, we take the square root of the discriminant D and hence if D is negative, its square root won't be a real number. The graph of such a quadratic equation does not touch or intersect the x-axis at any point.### Example

`x^2 + 6x + 5 = 0`**Calculate the discriminant:**`D = b^2 - 4ac`

`D = 6^2 - 4(1)(5)`

`D = 16`

**Use the table above to determine the nature of the roots:**Since D is greater than zero thus there are two real and distinct roots. Further, 16 is a

__perfect square__. Thus, the roots are__rational__in nature.**Compare the nature of roots to the actual roots:**Here is a graph of the above equation. It has two x-intercepts, -1 and -5, which are its roots or solutions. Notice that -1 and -5 are a pair of real, distinct and rational numbers. This coincides with our observation above the nature of roots above.