# Location of roots

Let quadratic equation

*ax*^{2}+*bx*+*c*= 0 has roots*Î±*and*Î²*. In some problems we want the roots of the equation to lie in a given interval. For this we impose conditions on*a*,*b*, and*c*.
Roots |
Conditions |

Î±, Î² > 0 |
Î± + Î² > 0, Î± â‹… Î² > 0, D â‰¥ 0 |

Î±, Î² < 0 |
Î± + Î² < 0, Î± â‹… Î² > 0, D â‰¥ 0 |

Î± < 0 < Î² |
Î± â‹… Î² < 0 |

Î±, Î² > k |
af(k) > 0, , D â‰¥ 0 |

Î±, Î² < k |
af(k) > 0, , D â‰¥ 0 |

Î± < k < Î² |
af(k) < 0 |

Î±, Î² âˆˆ (k_{1}, k_{2}) |
af(k_{1}) > 0, af(k_{2}) > 0, , D â‰¥ 0 |

Exactly one root says
Î± lies in (k_{1}, k_{2}) |
f(k_{1}) â‹… f(k_{2}) < 0 |

# Some important points

- Condition for general quadratic expression
*ax*^{2}+ 2*hxy*+*by*^{2}+2*gx*+ 2*fy*+*c*can be factorized into two linear factors is*abc*+ 2*fgh*â€“*af*^{2}â€“*bg*^{2}â€“*ch*^{2}= 0 and*h*^{2}â€“*ab*> 0. - If
*ax*^{2}+*bx*+*c*is perfect square then*b*^{2}â€“ 4*ac*= 0. - If
*Î±*is a repeated root of the quadratic equation*f*(*x*) =*ax*^{2}+*bx*+*c*= 0, then*Î±*is also a root of the equation*f*â€²(*x*) = 0. - If the ratio of roots of the quadratic equation
*ax*^{2}+*bx*+*c*= 0 be*p*:*q*, then*pqb*^{2}= (*p*+*q*)^{2}*ac*.