Being an avid Quora reader, I have found numerous answers that have intrigued me, connected with me, angered me, satisfied my thirst for information, and so on.
Given below is a quora answer given by a curious person with a good knowledge of mathematics. I have used the following excerpt to share my surprise with my friends and followers.
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SOURCE : https://qr.ae/pNrKPo
I bet that at some point in your life you have seen this formula:
This is the quadratic formula, which gives the solution to the quadratic equation:
To your wonder, there exists such formula for finding the roots of a cubic polynomial (polynomials having the highest power of variable xx as 3), too. But it is big and not recommended to learn, that’s probably why you never face it in the academic period.
- For the cubic equation: ax3+bx2+cx+dax3+bx2+cx+d where a≠0.a≠0.
The cubic formula to find the roots is:
- Now, about the fact that shocked me today, there is such formula for a polynomial of degree 4. Believe me, it is humongous as hell!
Let me show it to you.
Consider a quartic polynomial (i.e., polynomial with degree 4): ax4+bx3+cx2+dx+eax4+bx3+cx2+dx+e
Its solution (quartic formula) is:
The terms were grouped in the equation, which stops it from looking horrible. But it is very interesting and fascinating to see the entire formula in a single go.
- Here, the complete formula without dividing into chunks, for realizing its size.
For god’s sake, it is Gigantic!
Now you might be thinking that the size of such the solution to further polynomials of higher degree might go increasing. That’s not the case here.
That is, no such formula exists for polynomials having degree greater than or equal to 5. This is called the Abel-Ruffini theorem.
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Now let me share why this answer surprised me, the first reason was even though I have used the quadratic equation for years in school and college (Engineering Mathematics), I never had thought of it before. Also, it was awesome to know about the Abel-Ruffini theorem.
The second thought I had was if somehow in schools we were able to create the innate curiosity to find these in kids, our education system would be much better!
This reminds me of a shloka that I learned during my school times.
One fourth from the teacher, one fourth from own intelligence,
One fourth from classmates, and one fourth only with time.
AchAryAt pAdamAdatte, pAdam shiShyaH swamedhayA |
sa-brahmachAribhyaH pAdam, pAdam kAlakrameNa cha ||
आचार्यात् पादमादत्ते पादं शिष्यः स्वमेधया ।
सब्रह्मचारिभ्यः पादं पादं कालक्रमेण च ॥
This one shloka opens up many facets, related to learning and teaching — be it regular schools or spiritual learning. The modern education system, student-teacher relationship, undue blame on teachers or students for below expectation results.
Let us look at each of the four parts. one learns from four sources — teacher, self, others, and with time (i.e. experience).
This doesn’t mean it is exactly one-fourth from each. That may depend upon individual case s— how good the teacher, classmates, own intelligence, or environment is.
Just like to grow a tree a gardener, good seed, fertile soil, and time are needed, so too to learn one needs a teacher, own intelligence, co-students, and time (patience, experience).