Expanding and factorising quadratics

If you plot these graphs they may look the same, but are they?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Can you prove our inequality holds for all values of x and y between 0 and 1?

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

What do you get when you raise a quadratic to the power of a quadratic?