Chapter 1: The Oxford Math Challenge

Have you ever wondered if you possess the skills necessary to pursue mathematics at Oxford? I once doubted my own capabilities. However, through blogging, I've been able to assist aspiring students in navigating their way into prestigious mathematics programs.

To kick things off, here’s a thought-provoking question: what does it signify for two entities to intersect? I suggest you take a moment to pause, grab some writing materials, and give it a shot. Once you’re ready, continue for the solution!

Here we have two equations: one represents a straight line with slope (k), while the other depicts an upward-opening parabola. The slope (k) will change, and our task is to determine all the possible values for (k). Let’s substitute the linear equation into the parabolic one.

By expanding and rearranging the terms, we arrive at the quadratic equation:

[ x^2 - (k + 2)x + 1 = 0 ]

Can you follow the logic here? Essentially, the straight line will either merely touch the parabola at a single point or intersect it at two distinct points. This leads us to conclude that the discriminant of our quadratic must be greater than or equal to zero.

Recall the formula ( b^2 - 4ac ):

There are multiple approaches to tackle this. Here, we can utilize the difference of squares:

[

(k + 2 + 2)(k + 2 - 2) = (k + 4)(k) = 0

]

This means ( k = 0 ) or ( k = -4 ). You could even graph this for verification!

And there you have it — that’s the solution.

Photo by Kylie Osullivan on Unsplash

Isn’t that fascinating?

What were your thoughts during this exercise? Feel free to share in the comments, as I’d love to know!

Also, don’t forget to save and share the following curated list of top math puzzles available on Medium.

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