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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.

From the beginning of my schooling I had struggled with Mathematics. Highly discouraged by my results, I grew to eventually hate Maths and lost all motivation to improve. Then part way through year seven my parents decided to send me to Crystal Clear Mathematics. Graeme was able to quickly identify areas I needed to improve and explain Maths in a way that finally made sense to me. I was able to understand Maths. This in turn then affected my confidence in my own ability.

Since working with Graeme I have gone from failing to significantly improved results. I now get As & Bs both in examinations and for assignment tasks. Graeme is patient, kind and his tutoring is individualised. He was so helpful and I am grateful for his help.

Tessa M (student, 2014-2017)

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