Expanding and factorising quadratics

The sums of the squares of three related numbers is also a perfect square - can you explain why?

What is special about the difference between squares of numbers adjacent to multiples of three?

If you know the perimeter of a right angled triangle, what can you say about the area?

There are unexpected discoveries to be made about square numbers...

A 2-digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

This is a beautiful result involving a parabola and parallels.