Natures of the Roots of Quadratic Equations

Discriminant is the most important part of quadratic functions when we talk about the nature of its roots. Given a quadratic equation y=ax^2+bx+c, discriminant (D) can be defined mathematically as D=b^2-4ac.




Discriminant                                                   Nature of Roots

D=0                                                               Equal

D>0                                                               Real and distinct

D<0                                                               Imaginary Roots


Let me also show you the graph.

If D=0, it belongs to the family of perfect square trinomials in the form of y=a(x-h)^2 and has two the same roots.

If a<0

If a<0, Equal Roots



a>0. Equal Roots

If D>0, it belongs to the parabola that intersect the x-axis at two points

Real and Distinct root

Real and Distinct root

If D<0, it belongs to the parabola that the does not intersect the x-axis. That is why we call it imaginary because roots are the values of x that the given equation will intersect the x-axis. Does that make sense?


Imaginary roots

Imaginary roots

Worked Problem 1:

For what values of k does the equation 3x^2+2kx+3 to have equal roots?


For the equation to have real roots,


By substitution:

a=3,   b=2k,   c=3

(2k)^2-4(3)(3)=0 4k^2=36 k=\pm 3


Worked Problem 2: 2007 MMC Division finals

For what values of k does the equation y=3x^2+kx+3 has real and distinct roots?



By substitution:

a=3,   b=k,   c=3


If you don’t know how to solve this, better read first solving quadratic inequality.

Solving for k we have

k>6 or k<-6


Worked Problem 3:

For what values of k does the equation x^2+kx-1+k=0 will have an imaginary roots.



By substitution:

a=2,   b=k,   c=k-2

k^2-4(1)(k-1)<0 k^2-4k+4<0

k=2 with multiplicity of 2

If we follow the rules in quadratic inequality the solution must be 2<x<2 but is there an interval that is greater than 2 and less than 2? The answer is all real numbers except 2. Or we can simply say x\ne 2




Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

Latest posts by Dan (see all)

You may also like...

18 Responses

  1. Karri says:

    Ive by no means read something like this before. So good to find somebody with some original thoughts on this topic, really thank you for beginning this up. this web site is one thing that is necessary on the net, a person having a little originality. beneficial job for bringing some thing new to the web!

  2. Great blog! Do you have any recommendations for aspiring writers? I’m hoping to start my own blog soon but I’m a little lost on everything. Would you suggest starting with a free platform like WordPress or go for a paid option? There are so many options out there that I’m completely confused .. Any ideas? Bless you!

  3. this site says:

    C31EYg Just Browsing While I was surfing today I saw a excellent article concerning

  4. Chat says:

    356102 263665You completed certain very good points there. I did looking on the topic matter and identified most persons will go together along with your weblog 858884

  5. I every time used to read piece of writing in news papers but now as I am a user of internet therefore from now I am using net for articles
    or reviews, thanks to web.

  6. Im grateful for the blog. Fantastic.

  7. Oh my goodness! Awesome article dude! Many thanks, However I am
    having difficulties with your RSS. I don’t know the
    reason why I cannot subscribe to it. Is there anybody else getting similar
    RSS issues? Anybody who knows the solution can you kindly respond?

  8. May I simply just say what a comfort to discover someone who genuinely understands what they’re
    talking about online. You actually understand
    how to bring an issue to light and make it important.
    A lot more people really need to check this out
    and understand this side of the story. I can’t believe you’re not more popular given that you surely possess the gift.

  9. Greate article. Keep writing such kind of info on your site.
    Im really impressed by your blog.
    Hi there, You’ve done an incredible job. I’ll certainly digg it and for my part recommend to
    my friends. I am sure they’ll be benefited from this site.

  10. Tumblr says:

    136464 265985Its always good to learn tips like you share for blog posting. As I just started posting comments for blog and facing problem of lots of rejections. I think your suggestion would be helpful for me. I will let you know if its work for me too. 150220

  11. Dotty says:

    It is not my first time to pay a quick visit this web page,
    i am browsing this website dailly and take fastidious facts
    from here every day.

  12. We are a group of volunteers and opening a new scheme in our community.
    Your site offered us with valuable info to work on. You have done an impressive
    job and our whole community will be thankful to you.

  13. Attractive section of content. I simply stumbled upon your web site and
    in accession capital to claim that I get actually loved account your blog posts.

    Any way I’ll be subscribing in your augment or even I achievement you get admission to persistently quickly.

  14. Thanks for sharing your info. I really appreciate your efforts and I will be waiting for your further post thanks once again.

  15. Good post. I learn something totally new and challenging on sites I stumbleupon on a
    daily basis. It’s always interesting to read
    through content from other authors and use a little something from their sites.

  16. Katherine says:

    Hello terrific website! Does running a blog similar to this take a great deal of work?
    I have absolutely no knowledge of programming however I had been hoping to
    start my own blog soon. Anyways, should you have any
    ideas or techniques for new blog owners please share.
    I understand this is off topic but I just had to ask.
    Many thanks!

  17. hire amount says:

    This post is worth everyone’s attention. Where can I find out more?

  18. Wonderful beat ! I would like to apprentice at the
    same time as you amend your website, how could i
    subscribe for a weblog website? The account helped me a
    appropriate deal. I have been a little bit acquainted of
    this your broadcast provided brilliant clear concept

Leave a Reply

Your email address will not be published. Required fields are marked *