Quadratic equations

problem
Good approximations

Age

16 to 18

Challenge level

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
problem
Kissing

Age

16 to 18

Challenge level

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
problem
Two cubes

Age

14 to 16

Challenge level

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.]
problem
Golden thoughts

Age

14 to 16

Challenge level

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
article
Golden mathematics

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.
article
Continued fractions II

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
interactivity
Proof sorter - quadratic equation

Age

14 to 18

Challenge level

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
problem
A third of the area

Age

14 to 16

Challenge level

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?
problem
Mega quadratic equations

Age

14 to 18

Challenge level

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