Quadratic Equation Discriminant Proving Question

I realize many students have a challenge with presenting the solution of proving question. I am going to illustrate the correct way using the following question.

Show that y=x^2-x+\frac{3}{4} is always positive for all real values of x.

=> what is always +ve? the values of not x ( x can be of any values)

=> The graph will not touch x-axis; it is "hanging" above the x-axis

=> No roots, hence discriminant b^2-4ac is less than 0


  • Have an end in your mind. Be clear of the underlying concepts that the question is asking you to prove. Work towards that. For example, in the question, we want to prove that the discriminant b^2-4ac is less than 0. <== this is the end.
  • Sketch to have a clearer idea.


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Hi, I'm Ai Ling Ong. I enjoy coaching students who have challenges with understanding and scoring in 'O' Level A-Maths and E-Maths. I develop Math strategies, sometimes ridiculous ideas to help students in understanding abstract concepts the fast and memorable way. I write this blog to share with you the stuff I teach in my class, the common mistakes my students made, the 'way' to think, analyze... If you have found this blog post useful, please share it with your friends. I will really appreciate it! :)

6 Responses to Quadratic Equation Discriminant Proving Question

  1. I need help to solve the following ; Given tha Q is the range 180digrees <Q<270digrees and CCos Q=-40/41 and is angle in the range 90 degrees <& <180 Degrees such that sin & = 7/25
    a) Find the values of
    sin Q-tan&
    1-sin Q +cos &

    b) Solve all tthe angles of x i.e. between O degrees to 360 degrees for the equation sin Qx tan & = 1tanx


    alwaysLovely Reply:

    You would need to draw 2 triangles with one showing angle Q and the other showing angle &.
    Fill up the sides.
    These 2 triangles would be able to answer sin Q-tan&
    1-sin Q +cos &

    As for part (b), I do not quite get your question.


  2. show that 22/(3x²-4x+5) is positive for all real values of x and find its greatest value.


    Ai Ling Ong Reply:

    You can calculate the value of discriminant b^2 - 4ac.
    It should be less than 0 which means there is no real roots and the quadratic expression is always positive for all real values of x.
    To find its greatest value of 22/(3x²-4x+5), find the smallest value of 3x²-4x+5 by completing the square or finding stationary value of (3x²-4x+5).


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