Finding Quadratic Roots Geometrically or Graphically

Date: 12/07/2004 at 17:14:10
From: Adnan
Subject: Finding roots of quadratics with no real roots geometrically

Can somebody please tell me how to find the roots of a quadratic 
function geometrically?  For example, what is the algorithm to find 
roots for f(x) = x^2 + 1 by looking at the graph and "doing" things to 
it?  I read about it somewhere, but can not seem to remember where to 
find it again.  Thanks a bunch for the help.

Date: 12/07/2004 at 17:29:45
From: Doctor Schwa
Subject: Re: Finding roots of quadratics with no real roots geometrically

Hi Adnan,

If you know the coordinates of the vertex of the parabola--(0, 1) in
your example, or let's say (h,k) in general--then you can write

  f(x) = a*(x-h)^2 + k.

In order to make f(x) = 0, then, you need

  (x-h)^2 = -k/a,

so 

  x = h +/- sqrt(-k/a).

Amazingly enough, this turns out to be exactly the same as the
quadratic formula!  

Since h = -b/2a, that explains the first bit of the quadratic formula,
and k turns out to equal ... well, just the right thing to make this
expression the same as the quadratic formula.

In fact, finding the vertex of a parabola (by completing the square)
and proving the quadratic formula (by completing the square) are
almost the same thing!

Oh, and in case you're wondering how to find the value of 'a'
geometrically, you can do some things with slopes, or with the focus
and directrix if you want, or you can find the value when x = h+1 and
figure out 'a' quite quickly from that:

  f(h) = k,
  f(h + 1) = a + k,

so subtract those if you want to find out what 'a' is equal to.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/08/2004 at 08:49:30
From: Adnan
Subject: Thank you (Finding roots of quadratics with no real roots
geometrically)

Dr. Schwa,

Thank you very much for your reply (quite quick on top of that).  I
really appreciate it.  Take care.

Adnan