Home
Class 11
MATHS
If P(x) = ax^2+bx+c, and Q(x) = -ax^2+dx...

If `P(x) = ax^2+bx+c`, and `Q(x) = -ax^2+dx+c, acne `0, then prove that P(x).Q(x) =0 has at least two real roots.

Text Solution

Verified by Experts

`D_1` of `P(x) = b^2-4ac
`D_2 of Q(x)=d^2+4ac`
`(D_1+D_2)=b^2+d^2gt0`
Hence at least two roots of P(x) Q(x) = 0 are real.
Promotional Banner

Topper's Solved these Questions

  • QUADRATIC EQUATION

    PATHFINDER|Exercise QUESTION BANK|19 Videos

  • RECTANGULAR CO-ORDINATES

    PATHFINDER|Exercise QUESTION BANK|27 Videos

Similar Questions

Explore conceptually related problems

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If ax^(2)+ bx +c and bx ^(2) + ax + c have a common factor x +1 then show that c=0 and a =b.

If 2a+3b+6c=0 " "(a, b, c in R) , then the quadratic equation ax^(2)+bx+c=0 has at least one root -

If a, b , c in R and 3b^(2)-8ac lt 0 , then the equation ax^(4)+bx^(3)+cx^(2)+5x-7=0 has a) all real roots b) all imaginary roots c)exactly two real and two imaginary roots d) none

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

Let f(x)=ax^(2)+bx+c , g(x)=ax^(2)+qx+r , where a , b , c , q , r in R and a lt 0 . If alpha , beta are the roots of f(x)=0 and alpha+delta , beta+delta are the roots of g(x)=0 , then

If the equation a x^2+b x+c=x has no real roots, then the equation a(a x^2+b x+c)^2+b(a x^2+b x+c)+c=x will have a. four real roots b. no real root c. at least two least roots d. none of these

If 2a+3b+6c=0 , show that the equation ax^(2)+bx+c=0 has at least one root between 0 and 1.

If alpha and beta are the roots of ax^2+bx+c =0, then the equation ax^2-bx(x-1)+c(x-1)^2 =0 has roots

PATHFINDER-QUADRATIC EQUATION AND EXPRESSION-QUESTION BANK
  1. If (cgt 0) and 2ax^2+3bx+5c=0 does not have any real roots, then prove...

    Text Solution

    |

  2. If 2x^3 + ax^2+bx+4 =0 (a and b are positive real numbers) has 3 real ...

    Text Solution

    |

  3. If P(x) = ax^2+bx+c, and Q(x) = -ax^2+dx+c, acne 0, then prove that P(...

    Text Solution

    |

  4. If alpha,beta,gaama are the roots of the cubic x^3+qx+r=0, find the eq...

    Text Solution

    |

  5. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  6. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  7. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  8. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  9. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  10. f(x)=x^2-(m-3)x+m=0 is a quadratic equation, find values of m for whic...

    Text Solution

    |

  11. If,alpha be the root of equation ax^2+bx+c=0 and beta be root of -ax^2...

    Text Solution

    |

  12. Find the values of 'm' for which the equation x^4-(m-3)x^2+m=0 has Fo...

    Text Solution

    |

  13. Find the values of 'm' for which the equation x^4-(m-3)x^2+m=0 has Fo...

    Text Solution

    |

  14. Find the values of 'm' for which the equation x^4-(m-3)x^2+m=0 has No...

    Text Solution

    |

  15. For what value of 'm' inR, both roots of the equation x^2-6mx+9m^2-2m+...

    Text Solution

    |

  16. Find the value(s) of 'a' for which the inequality tan^(2)x+(a+1) t...

    Text Solution

    |

  17. Find the common root of x^2-3x+2=0 and x^2+x-2=0

    Text Solution

    |

  18. Find the condition if equations ax^2+bx+c=0 and x^2+2x+3=0 have a comm...

    Text Solution

    |

  19. If x^2+ax+bc=0 and x^2+bx+ac=0 have a common root, show their other ro...

    Text Solution

    |

  20. If alpha,beta are the roots x^2+px+q=0 and gamma,delta are the roots o...

    Text Solution

    |