Vieta’s Formula in Quadratic Equations

Given a quadratic equation f(x)=ax^2+bx+c, let and q be the roots of this equation then, we can relate the sum of the roots as well as the product of the roots by the following formula. This is called Viete’s formula.


p+q=\displaystyle\frac{-b}{a}
pq=\displaystyle\frac{c}{a}

The following formula was discovered by François Viète. The formula is not just for quadratic but for other polynomial equations with higher degree but we will just deal with quadratic equations for now.

 

Worked Problem 1:
Let p and q be the roots of equation 2x^2-x+4=0, what is the value of \displaystyle\frac{2}{q} +\displaystyle\frac{2}{p}

Solution:
Using Viete’s formula, we know that

p+q=\displaystyle\frac{1}{2}
pq= 2

We are asked to find the value of the following

\displaystyle\frac{2}{q} +\displaystyle\frac{2}{p}
=2(\displaystyle\frac{1}{q} +\displaystyle\frac{1}{p}) by common factor
=2(\displaystyle\frac{p+q}{pq})
=2(\displaystyle\frac{\frac{1}{2}}{2})
=\displaystyle\frac{1}{2}

Worked Problem 2:

Let a and b be the roots of equation x^2-3x+1. Find the value of a^3+b^3.

Solution:
By Viete’s formula,

a+b=3 and ab=1

We are asked to find the value of a^3+b^3

a^3+b^3=(a+b)(a^2-ab+b^2) by factoring special product.

We already have (a+b) we just to find a^2+b^2

(a+b)^2=a^2+2ab+b^2
a^2+b^2=(a+b)^2-2ab

Going back here

a^3+b^3=(a+b)((a+b)^2-2ab-ab) since a^2+b^2=(a+b)^2-2ab
a^3+b^3=(a+b)((a+b)^2-3ab)
a^3+b^3=(3)(3^2-3(1))
a^3+b^3=18

Worked Problem 3:

Find the value of K such that the sum of the roots of quadratic kx^2+(k+1)x+1=0 is -2.

Solution:
Let p and q are the roots of equation.
By Vieta’s formula,

p+q=\displaystyle\frac{-b}{a}
-2=\displaystyle\frac{(k+1}{-k}
2k=k+1
k=1

Dan

Dan

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.
Dan

Latest posts by Dan (see all)

You may also like...

14 Responses

  1. Anonymous says:

    sir dan wala ba sa cubic , quartic or even a general Vieta’s formula for all types of degree 🙂

  2. Dan Lang says:

    We will discuss that shortly.

  3. Rafael says:

    I was just searching for this information for some time. After six hours of continuous Googleing, at last I got it in your site. I wonder what’s the lack of Google strategy that don’t rank this kind of informative websites in top of the list. Usually the top websites are full of garbage.

  4. Pretty section of content. I just stumbled upon your website and in accession capital to assert that I get in fact enjoyed account your blog posts. Any way I’ll be subscribing to your augment and even I achievement you access consistently quickly.

  5. seo says:

    Hello Web Admin, I noticed that your On-Page SEO is is missing a few factors, for one you do not use all three H tags in your post, also I notice that you are not using bold or italics properly in your SEO optimization. On-Page SEO means more now than ever since the new Google update: Panda. No longer are backlinks and simply pinging or sending out a RSS feed the key to getting Google PageRank or Alexa Rankings, You now NEED On-Page SEO. So what is good On-Page SEO?First your keyword must appear in the title.Then it must appear in the URL.You have to optimize your keyword and make sure that it has a nice keyword density of 3-5% in your article with relevant LSI (Latent Semantic Indexing). Then you should spread all H1,H2,H3 tags in your article.Your Keyword should appear in your first paragraph and in the last sentence of the page. You should have relevant usage of Bold and italics of your keyword.There should be one internal link to a page on your blog and you should have one image with an alt tag that has your keyword….wait there’s even more Now what if i told you there was a simple WordPress plugin that does all the On-Page SEO, and automatically for you? That’s right AUTOMATICALLY, just watch this 4minute video for more information at. Seo Plugin

  6. I just got these currently and like, really like, adore them! I’ve been in search of mongolian sheep human hair weave https://www.youtube.com/watch?v=koiFnDsfNPU,glad I chose these. Won’t be able to wait to put on them yet again! I am hooked!

  7. fd1dkJ with hackers and I am looking at alternatives for another platform. I would be great if you could point me in the direction of a good platform.

  8. 557363 405787Hey this is kinda of off topic but I was wanting to know if blogs use WYSIWYG editors or if you have to manually code with HTML. Im starting a blog soon but have no coding experience so I wanted to get advice from someone with experience. Any help would be greatly appreciated! 55015

  9. 136624 998103I would like to see far more posts like this!.. Wonderful weblog btw! reis Subscribed.. 785940

  10. Can I just say what a relief to find someone who actually knows what theyre talking about on the internet. You definitely know how to bring an issue to light and make it important. More people need to read this and understand this side of the story. I cant believe youre not more popular because you definitely have the gift.

  11. Ahaa, its fastidious discussion about this paragraph at this
    place at this webpage, I havve read all that, so at this time me also commenting here.

  12. Färger says:

    You really make it seem so easy with your presentation but I find this topic to be actually something that I think I would never understand. It seems too complicated and extremely broad for me. I am looking forward for your next post, I will try to get the hang of it!

  13. I do not even know how I ended up here, but I thought this post was good. I do not know who you are but certainly you are going to a famous blogger if you are not already 😉 Cheers!

  14. Mopperace says:

    Wow, incredible blog layout! How long have you been blogging for? you make blogging look easy. The overall look of your site is fantastic, let alone the content!

Leave a Reply

Your email address will not be published.