Today’s post will be fairly short and simple, but may have interesting usages in other areas. Given a quadratic equation , it is well known that solutions for are given by the quadratic formula, . There are a multitude of ways to derive this:

• Completing the square: can be reduced to . At this point, we know easily how to solve equations of the form , and by extension , by simply taking square roots (being wary of the two signs) and finishing from there. Applying some wishful thinking shows that our general quadratic can be reduced to an equation of exactly that form:
  

     

  From here, simply moving the constant term to the other side, taking square roots and subtracting yields the quadratic formula.


• Many of you may have come across this post by Po-Shen Loh detailing an interesting approach to solving quadratics. Suppose we just consider quadratics with leading coefficient , as we have shown that any quadratic can be reduced to this by dividing out the leading coefficient. By the Fundamental Theorem of Algebra, this quadratic can be factored in terms of its roots and :
  

     

  By matching coefficients, we have and . We note that the average of and equals because their sum is , so we substitute and with and respectively, for some . From here, we may use the second piece of information regarding the roots; that their product is . Thus we have . At this point, we can clearly solve for by isolating it, and consequently solve for and thus both roots.

I noticed that there is another method by which quadratics can be solved that follows along with the themes of the last approach; the usage of Vieta’s formulas. Given information about and , how may we isolate and ? Notice that there is an important identity involving , and : . We know the left hand side to be , and we can similarly find on the right side, but this doesn’t help much for isolating and . But wait, there is another important identity involving very similar terms: . From here, we note that taking and subtracting from both sides produces the second identity on the right side. We have

   

which is a very good sign because we can calculate the LHS by the coefficients of the quadratic. Taking the square root, we can directly calculate . But now that we know both and , we can use simple elimination techniques for solving linear equations to isolate each root, such as addition and subtraction of the equations. Note that we do not need to worry about the two possible square roots because if we choose the negative option, the values of and will simply alternate. I hope similar methods involving Vieta’s formulas may be generalized to higher degree polynomials, such as cubics.