tag:blogger.com,1999:blog-3722233.post5076831303121480999..comments2023-03-25T10:00:22.914-05:00Comments on Computational Complexity: The Quadratic FormulaLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-3722233.post-7332716065285235652009-05-19T12:50:00.000-05:002009-05-19T12:50:00.000-05:001. In reply to anonymous 2
It is not always the ca...1. In reply to anonymous 2<br />It is not always the case that proofs are essential. What is more useful: Chernoff's formula or knowing how to prove it?<br /><br />2. If I were to teach the derivation of the quadratic formula for children, I would first teach coordinate changes on the plane, then reduce the equation to the form<br />x^2 + b/a x + c/a = 0<br />(emphasizing we must have "a" nonzero),<br />then solve the simpler family of equations without the linear term, and finally look for a coordinate transformation in which the linear term disappears.<br /><br />The advantage of this approach is that, although it is slow and complicated, it illustrates a general plan to solve problems by decomposing it into simpler tasks, and, unlike just "completing the square", each task is a useful skill, with more applications.<br /><br />Of course, using Wolfram may be equally useful.CSProfhttps://www.blogger.com/profile/07212822875614144307noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-69770310087934056892009-05-19T08:18:00.000-05:002009-05-19T08:18:00.000-05:00Wolfram alpha shows how to do it if you ask it to ...Wolfram alpha shows how to do it if you ask it to "solve ax^2+bx+c=0" (click show steps):<br />http://www79.wolframalpha.com/input/?i=solve+ax%5E2%2Bbx%2Bc%3D0noamhttp://agtb.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-29425408595803153122009-05-17T19:11:00.000-05:002009-05-17T19:11:00.000-05:00Off topic, but related to TCS. There is a posting...Off topic, but related to TCS. There is a posting for stimulus postdocs at the CRA site.<br /><br />It's great that they have this, but why is the salary so high? Do they realize that many people who got their PhD's 2+ years ago (and are therefore ineligible) still do not have jobs, and yet they are giving extremely high salary (for a postdoc) jobs ($75,000 with $25,000 in "fringe benefits" and $15,000 per year "discretionary" cost for travel, etc) rather than making more jobs and employing more people.<br /><br />Makes me wish there had been a financial crisis when I was graduating ..Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-67616523438810628662009-05-16T01:34:00.000-05:002009-05-16T01:34:00.000-05:00Completing the square:
ax^2 + bx + c = 0
=>...Completing the square:<br /><br /> ax^2 + bx + c = 0<br />=> x^2 + (b/a)x + (c/a) = 0 <br /><br /> (here you need a != 0)<br /><br />=> x^2 + (b/a)x = -(c/a)<br /><br />Now we want to make the left hand side a square -- we want to be able to write it in the form (x+y)^2.<br /><br />But (x+y)^2 = x^2 + 2xy + y^2 , so we go:<br /> 2xy = (b/a)x <br />=> y = (b/2a)<br /><br />Now we add y^2 to both sides:<br /><br />x^2 + (b/a)x + (b^2/4a^2) = (b^2/4a^2) -(c/a)<br />=> (x+ (b/2a))^2 = (b^2 - 4ac)/4a^2<br /><br />Taking square roots on both sides,<br /><br /> x + (b/2a) = +- \sqrt{(b^2 - 4ac)}/2a<br /><br />, and so<br /><br /> x = (-b +- \sqrt{(b^2 - 4ac)})/2aPhilipnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-28985407046131152422009-05-15T12:15:00.000-05:002009-05-15T12:15:00.000-05:00Did you find out?
It is pretty simple:
(x-x1)*(x-x...Did you find out?<br />It is pretty simple:<br />(x-x1)*(x-x2)=0<br />now x1+x2=-b/a x1*x2=c/a<br />now let x1 = m-n, x2 = m+n (m=(x1+x2)/2)<br />so x1+x2 = 2m = -b/a<br />x1*x2=m^2-n^2=c/a<br />now m = -b/2a<br />and b^2/4a^2 - c/a = n^2.<br />So we have n^2 = (b^2-4ac)/4a^2<br />or n = sqrt(b^2-4ac)/2a and -sqrt(b^2-4ac)/2a<br />now x1 = m-n, x2 = m+n or<br />x1 = (-b - sqrt(...))/2a <br />and x2 = (-b + sqrt(...))/2a.Unknownhttps://www.blogger.com/profile/00153875255110809530noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-79916502719928110822009-05-15T10:41:00.000-05:002009-05-15T10:41:00.000-05:00Teaching how to solve second order equations witho...Teaching how to solve second order equations without giving the proof of the formula is useless! Blindly following 'recipes' has no benefit to students!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-37997518480008268842009-05-15T08:51:00.000-05:002009-05-15T08:51:00.000-05:00You can always use Monte Carlo methods...You can always use Monte Carlo methods...Helgerhttps://www.blogger.com/profile/16943917820293652157noreply@blogger.com