# Roots of a Quadratic Equation and their Nature

Given is the quadratic equation

*ax*^{2}+*bx*+*c*= 0, where a â‰ 0.Then

It can be seen that the nature of both these two roots depends upon the value of

*b*^{2}â€“ 4*ac*.(

*b*^{2}â€“ 4*ac*) is also known as**Discriminant (D)**of the quadratic equation*ax*^{2}+*bx*+*c*= 0.If D = 0, then

So, the roots will be real and equal.

If D > 0, then

So, the roots will be real and distinct.

If D < 0, then is not real.

So, the roots will not be real.

If D is a perfect square (including D = 0) and

So, there are seven set of values for (

*a*,*b*and c are rational, then the roots will also be rational.Example

Given is the quadratic equation

*ax*^{2}+*bx*+ 1 = 0, where*a*,*b*(1, 2, 3, 4). For how many set of values of (*a*,*b*), the quadratic equation*ax*^{2}+*bx*+ 1 = 0 will have real roots?**(CAT 2003)**Solution

For the roots to be real, D â‰¥ 0.

D =

*b*^{2}â€“ 4*a*â‰¥ 0.Forming the table for the above-written condition:

b |
a |

1 | Not possible |

2 | 1 |

3 | 1 |

3 | 2 |

4 | 1 |

4 | 2 |

4 | 3 |

4 | 4 |

*a*,

*b*).