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More Advanced Quadratic Problems

Three advanced quadratic equations involving trigonometry, exponentials and quartic functionsSometimes other functions are embedded “inside” quadratic equations.

When this occurs, we normally solve the quadratic equation first, and then deal with each of the two resulting equations.  At first, we learn to do this using substitution so as to avoid confusion.  After a while, when the patterns become more clear, we can dispense with substitution and simply manipulate the equation as it is.



Using Squares of Binomials to Solve a Difficult Problem ... a² + b² + c² = ab + bc + ca

When I was about 15, I was given a flier that was advertising an IBM-sponsored Mathematics Competition. One of the sample questions provided was this one:

The real numbers a, b, and c, are such that a² + b² + c² = ab + bc + ca.  Prove that a = b = c.

This looks like an extremely difficult problem ... but I wish to share that it is not as intractable as it appears!

In this short video I explain the principles involved and how we can, indeed, conclude that the three values must all be the same. Even if you feel that you are not particularly good with mathematics, why not watch the video and see if you can follow the flow of the argument? I hope you will find it encouraging.

And here is a scan of the flier that I was given. It inspired me to enter the competition.

IBM Mathematics Competition Flier for 1970

And if you would like to see the test that I sat for back then, here is a PDF copy of the 1970 IBM Mathematics Competition for NSW.

Hi! I have received your message and checked this video. Your methods you mention for this video is brilliant! I am starting it now. … Thank you very much again for this video and I am hoping to watch more!
Shinyfio (on a CCM YouTube video about Using a Diary to Help Your Long Term Memory)

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