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
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