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.

Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise All Questions|363 Videos
  • CONIC SECTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1167 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

CENGAGE PUBLICATION-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-All Questions
  1. Find the quadratic equation with rational coefficients whose one root ...

    Text Solution

    |

  2. Let z be a complex number satisfying the equation (z^3+3)^2=-16 , then...

    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. Find the real numbers x and y if (x-iy)(3+5i) is the conjugate of -6-2...

    Text Solution

    |

  5. If x is real, then x//(x^2-5x+9) lies between a.-1a n d-1//11 b. ...

    Text Solution

    |

  6. Find the least positive integer n such that ((2i)/(1+i))^n is a positi...

    Text Solution

    |

  7. Set of all real value of a such that f(x)=((2a-1)x^2+2(a+1)x+(2a-1))/(...

    Text Solution

    |

  8. Find the real part of e^(ei theta)

    Text Solution

    |

  9. If alpha, beta and gamma are the roots of x^3 - x^2-1 = 0, then value ...

    Text Solution

    |

  10. Prove that z=i^i, where i=sqrt-1, is purely real.

    Text Solution

    |

  11. If alpha,beta,gamma,delta are the roots of the equation x^4-K x^3+K x^...

    Text Solution

    |

  12. In A B C ,A(z1),B(z2),a n dC(z3) are inscribed in the circle |z|=5. I...

    Text Solution

    |

  13. Suppose that f(x) isa quadratic expresson positive for all real xdot I...

    Text Solution

    |

  14. Multiply: (2+5i)(4-3i)

    Text Solution

    |

  15. Let f(x)=ax^2-bx+c^2, b != 0 and f(x) != 0 for all x ∈ R. Then (a) ...

    Text Solution

    |

  16. It is given that n is an odd integer greater than 3 but n is not a mul...

    Text Solution

    |

  17. If a ,b in R ,a!=0 and the quadratic equation a x^2-b x+1=0 has imagi...

    Text Solution

    |

  18. Find the complex number omega satisfying the equation z^3=8i and lyi...

    Text Solution

    |

  19. If the expression [m x-1+(1//x)] is non-negative for all positive real...

    Text Solution

    |

  20. When the polynomial 5x^3+M x+N is divided by x^2+x+1, the remainder is...

    Text Solution

    |