Quadratic equations

Exploit the symmetry and turn this quartic into a quadratic.

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

What does Pythagoras' Theorem tell you about the radius of these circles?

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

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.]

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

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?

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

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!