Roots of Quartic Equation

In high school, we learned that the roots of the quadratic equation ax^2 + bx + c = 0 are given by

\displaystyle r_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

and

\displaystyle r_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}.

Last week, a student asked me whether a similar formula exists for roots of a quartic equation. A quartic equation is an equation of degree four, which has four roots (a quadratic equation is of degree two, and therefore has two roots).

I told him yes, there is such formula, but there’s a reason why it is not taught in high school.

The roots r_1, r_2, r_3, r_4 of the quartic equation x^4 + ax^3 + bx^2 + cx + d = 0 are given by the formula

(click on the image to enlarge)

No such formula exists for equations with degree 5 or higher, a fact that was proven by Norwegian mathematician Neils Henrik Abel in 1824 when he was only 22 years old (he discovered a proof three years earlier, but it was later found to contain an error). This theorem is known today as the Abel-Ruffini theorem.

12 Responses to Roots of Quartic Equation

  1. yus rezal says:

    Bal, kenapa ko terer math ye bal.apa yang ko nampak disebalik nombor² dan huruf² tu? apsal aku faham sains tapi tak paham soalan math?- (sekadar melepas kan tension yg mana semalam exam basic math aku tak dapat jawab,hampeh!)

  2. suhaimiramly says:

    kekeke aku takde lah terer mana. basically its a two-step process:

    1. identify your unique natural talent — math, sports, drawing, piano, poetry, singing, design, writing, gitar, main bola, investing, memasak, programming etc etc. whatever it is

    2. practice on it for 10,000 hours (roughly 10 years at 3 hours a day)

    nanti jadi terer lah

    (ni aku baca dalam buku “Outliers” by Malcolm Gladwell)

    mostly people do not work on their natural talent tapi mereka buat benda yang didesak oleh keluarga/rakan sebaya dan dianggap ‘normal’ dalam masyarakat (study engineering, for example) walaupun mereka tak minat/tak berbakat dalam bidang itu. sebab tu lah jadi average je, tak jadi hebat

  3. yus rezal says:

    dulu dekan engineering aku kat UTM pun ada bg analogi yg lebih kurang sama.dia kata kalo pandai dan suka main bola,jgn la wakil uni main hoki plak.train la macam mana pun,jadi biasa-biasa je la.so, kena configure natural talent dulu la.heh heh heh

  4. suhaimiramly says:

    agree with him.

    tapi untuk melalui benda ni, ibubapa juga perlu mainkan peranan:
    1. identify kebolehan anak seawal mungkin
    2. nurture bakat tersebut
    3. jangan halang mereka dari pursue bakat tersebut sehingga ke peringkat tertinggi.

    No 3 tu paling susah sekali. ramai parents suruh jugak anak dia belajar medic & jadi doktor walaupun anak dia tak minat. kadang2 aku sedih tengok some kids who are math genius, tapi terpaksa buat medic and then later have to drop out sebab tak minat.

    Dulu masa lepas SPM my parents also beriya suruh aku buat medic. nasib baik they are reasonable, i explained to them that studying math is also useful. and i feed them bullshit about sapa terer math boleh jadi actuary and make $$$ (padahal aku taknak jadi actuary pun). tapi at least mereka faham. i wouldnt survive 2 years in medical school — im not cut out for that.

    jadi parents2 sekalian — JANGAN paksa anak buat benda2 yang dia tak minat. nanti akan destroy masa depan anak tu, dan akan wujud love-hate relationship between you and the child.

    • yus rezal says:

      Pergh! aku mmg setuju yang nih.aku ada live experience.bukan terjadi pada aku la.tp org yg rapat ngan aku.lambat laun bila kita sedar,kita mula rasa ada yang tak kena dgn hidup,lepas tu dah tentu ada love-hate relationship tu.dan parent tu lebih senang menjadi “who to blame” satu hari nnt.

      • lyndaz says:

        huhu~
        aku bru jer stat klas bio td utk psdiaan medic.
        T.T …aku mmg sgt xmnat bio!
        tp xda options r..parents sroh amik medic.
        seyesly r.aku love giller2 math n addm3.
        even rzult aku pon(spm 2009)
        m3 n addm3 lg best byk dri bio.
        haha~
        sgt trasa nk tkar cos neh.tp..
        tpksa lak say bye2 kt sholarship!

  5. suhaimiramly says:

    but it raises a question, camana kalau bakat anak tu bukan akademik, contoh bakat main bola, lari pecut, main alat music, melukis etc?

    patutkah anak itu teruskan pursue bakat tersebut, atau perlukah dia masuk universiti belajar macam orang lain?

    what do you guys think

  6. yus rezal says:

    Pada pendapat aku kena masuk U jugak Bal.tp still pursue apa yg dia nak jadi la kan.Education penting,nak cari keje orang tgk sijil,diploma,degree.nak keje sendiri modal nak ada,pengalaman kena ada.nak cari pengalaman kena kerja dulu.kalo main bola ke,tenis ke,musician ke,kena excel dari awal.tahap wakil negara ke (cth Nicol David),baru la berbaloi.kalo main bola tahap negeri pun, sekali patah kaki,pastu takleh guna lagi.menjawab beb.anak bini sapa mau tanggung?kena la kerja…sekadar pendapat yang tah apa apa…..

    • suhaimiramly says:

      tul jugak. aku tengok ex-athlete yang dah retire kebanyakan hidup miskin walaupun masa muda dulu mereka juara / pemenang pingat macam2. mungkin lepas mereka bersara mereka tak ada marketable skills except jadi coach. itu pun berapa kerat je kerja coach yang available.

      yg kena retire sebab injury lagi kesian…dulu kan ada bersamamu tv3 yg khas untuk former athletes

      kesimpulannya: belajar sampai abis

  7. samah says:

    hahaha bal.. teringat ko dok citer pasal paper2 yang kene amik untuk actuarial science masa kat ppp dulu…

  8. suhaimiramly says:

    itu semua bullshit kekkeekeke…aku tak penah minat pon bidang actuary ni…sampai skarang aku tak tau apa benda keja actuary nih

    • atoq says:

      Love your banter and insight, made me think twice about some of the things I forced my children to take take up to prepare them for their future. Maybe I should invest more energy trying to find out what is it that makes them tick right now. Thanks.

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